金声玉亮2.0模型评测报告

问题

You are given a question and its solution. The solution however has its formulae masked out using the tag <missing X> where X indicates the identifier for the missing tag. You are also given a list of formulae in latex in the format "<expression Y> = $<latex code>$" where Y is the identifier for the formula. Your task is to match the formulae to the missing tags in the solution. Think step by step out loud as to what the answer should be. If you are not sure, give your best guess. Your answer should be in the form of a list of numbers, e.g., 5, 22, 3, ..., corresponding to the expression identifiers that fill the missing parts. For example, if your answer starts as 5, 22, 3, ..., then that means expression 5 fills <missing 1>, expression 22 fills <missing 2>, and expression 3 fills <missing 3>.

The question is:
Find all integers $n \geq 3$ such that the following property holds: if we list the divisors of $n !$ in increasing order as $1=d_1<d_2<\cdots<d_k=n!$, then we have \[ d_2-d_1 \leq d_3-d_2 \leq \cdots \leq d_k-d_{k-1} .\]
-Find all integers $n \geq 3$ such that the following property holds: if we list the divisors of $n !$ in increasing order as $1=d_1<d_2<\cdots<d_k=n!$, then we have \[ d_2-d_1 \leq d_3-d_2 \leq \cdots \leq d_k-d_{k-1} .\]

The solution is:
We can start by verifying that $n=3$ and $n=4$ work by listing out the factors of $3!$ and $4!$ . We can also see that <missing 1> does not work because the terms $15, 20$ , and $24$ are consecutive factors of <missing 2> . Also, <missing 3> does not work because the terms $6, 8$ , and $9$ appear consecutively in the factors of $6!$ .
-We can start by verifying that $n=3$ and <missing 4> work by listing out the factors of $3!$ and $4!$ . We can also see that $n=5$ does not work because the terms <missing 5> , and $24$ are consecutive factors of $5!$ . Also, <missing 6> does not work because the terms <missing 7> , and $9$ appear consecutively in the factors of <missing 8> .

Note that if we have a prime number <missing 9> and an integer <missing 10> such that both $k$ and <missing 11> are factors of $n!$ , then the condition cannot be satisfied.

If <missing 12> is odd, then $(2)(\frac{n-1}{2})(n-1)=n^2-2n+1$ is a factor of $n!$ . Also, <missing 13> is a factor of <missing 14> . Since <missing 15> for all $n\geq7$ , we can use Bertrand's Postulate to show that there is at least one prime number $p$ such that <missing 16> . Since we have two consecutive factors of $n!$ and a prime number between the smaller of these factors and $n$ , the condition will not be satisfied for all odd <missing 17> .

If <missing 18> is even, then <missing 19> is a factor of $n!$ . Also, $(n-3)(n-1)=n^2-4n+3$ is a factor of $n!$ . Since $2n<n^2-4n+3$ for all $n\geq8$ , we can use Bertrand's Postulate again to show that there is at least one prime number $p$ such that $n<p<n^2-4n+3$ . Since we have two consecutive factors of <missing 20> and a prime number between the smaller of these factors and $n$ , the condition will not be satisfied for all even <missing 21> .

Therefore, the only numbers that work are <missing 22> and <missing 23> .

The formulae are:
<expression 1> n=5
<expression 2> 15, 20
<expression 3> n\geq7
<expression 4> n=6
<expression 5> n=4
<expression 6> n!
<expression 7> 6, 8
<expression 8> n<p<n^2-2n
<expression 9> p>n
<expression 10> k>p
<expression 11> n=3
<expression 12> (n-2)(n)=n^2-2n
<expression 13> n!
<expression 14> 5!
<expression 15> k+1
<expression 16> (2)(\frac{n-2}{2})(n-2)=n^2-4n+4
<expression 17> n=6
<expression 18> n\geq8
<expression 19> n=4
<expression 20> 6!
<expression 21> n\geq8
<expression 22> 2n<n^2-2n
<expression 23> n\geq7

Your final answer should be STRICTLY in the format:

<Detailed reasoning>

Answer: <comma separated list of numbers representing expression identifiers>

答案

模型	模型答案	正确答案
金声玉亮2.0	<Detailed reasoning> To match the formulae to the missing tags in the solution, we need to carefully analyze the context in which each <missing X> appears and find the corresponding expression from the list. Here's the step-by-step reasoning: 1. <missing 1>: The solution mentions that a certain value of $n$ does not work due to specific terms being consecutive factors. The context suggests it's about a specific $n$ value, and the only expression directly mentioning a specific $n$ value in this context is <expression 1> $n=5$. 2. <missing 2>: This refers to the factors of a specific $n!$, and the context mentions $15, 20, 24$ as consecutive factors. The only expression listing these terms is <expression 2> $15, 20$. 3. <missing 3>: Similar to <missing 1>, this refers to a specific $n$ value that does not work. The context mentions $n=6$, which corresponds to <expression 4> $n=6$. 4. <missing 4>: This refers to a specific $n$ value that works, and the context mentions $n=4$, which corresponds to <expression 5> $n=4$. 5. <missing 5>: This refers to the terms $15, 20$ being consecutive factors of $5!$, which corresponds to <expression 2> $15, 20$. 6. <missing 6>: This refers to a specific $n$ value that does not work, and the context mentions $n=6$, which corresponds to <expression 4> $n=6$. 7. <missing 7>: This refers to the terms $6, 8$ being consecutive factors of $6!$, which corresponds to <expression 7> $6, 8$. 8. <missing 8>: This refers to the factors of $6!$, which corresponds to <expression 20> $6!$. 9. <missing 9>: This refers to a prime number $p$ greater than $n$, which corresponds to <expression 9> $p>n$. 10. <missing 10>: This refers to an integer $k$ such that $k>p$, which corresponds to <expression 10> $k>p$. 11. <missing 11>: This refers to $k+1$, which corresponds to <expression 15> $k+1$. 12. <missing 12>: This refers to an odd $n$, and the context suggests it's about a specific condition for odd $n$, which corresponds to <expression 3> $n\geq7$. 13. <missing 13>: This refers to a factor of $n!$, which corresponds to <expression 13> $n!$. 14. <missing 14>: This refers to a specific $n!$, and the context mentions $5!$, which corresponds to <expression 14> $5!$. 15. <missing 15>: This refers to a condition for all $n\geq7$, which corresponds to <expression 22> $2n<n^2-2n$. 16. <missing 16>: This refers to a prime number $p$ such that $n<p<n^2-2n$, which corresponds to <expression 8> $n<p<n^2-2n$. 17. <missing 17>: This refers to all odd $n\geq7$, which corresponds to <expression 3> $n\geq7$. 18. <missing 18>: This refers to an even $n$, and the context suggests it's about a specific condition for even $n$, which corresponds to <expression 18> $n\geq8$. 19. <missing 19>: This refers to a specific even $n$ value, and the context mentions $n=4$, which corresponds to <expression 5> $n=4$. 20. <missing 20>: This refers to the factors of $n!$, which corresponds to <expression 13> $n!$. 21. <missing 21>: This refers to all even $n\geq8$, which corresponds to <expression 21> $n\geq8$. 22. <missing 22>: This refers to a specific $n$ value that works, and the context mentions $n=3$, which corresponds to <expression 11> $n=3$. 23. <missing 23>: This refers to a specific $n$ value that works, and the context mentions $n=4$, which corresponds to <expression 5> $n=4$. Answer: 1, 2, 4, 5, 2, 4, 7, 20, 9, 10, 15, 3, 13, 14, 22, 8, 3, 18, 5, 13, 21, 11, 5	1,14,17,19,2,4,7,20,9,10,15,23,12,6,22,8,3,18,16,13,21,11,5
deepseek-r1	<Detailed reasoning> The solution checks small n values first. For missing1, the example with terms 15,20,24 in 5! implies n=5 (expression1). missing2 is 5! (expression14). For n=6 not working (missing3), use expression4. missing4 refers to n=4 (expression5). missing5 lists 15,20 (expression2). missing6 is n=6 (expression4), and missing7 is 6,8 (expression7). missing8 is 6! (expression20). For primes, missing9 is p>n (expression9), missing10 is k>p (expression10), and missing11 is n=3 (expression11). When n is odd, missing12 is the formula (expression12). missing13 is n! (expression13), missing14 is 5! (expression14). missing15 refers to n≥7 (expression3). Bertrand's interval is expression8. missing17 is n≥7 (expression3). For even n≥8 (expression18), missing19 is the formula (expression16). missing20 is n! (expression6), and missing21 is n≥8 (expression18). Finally, the valid n's are 3 (expression11) and 4 (expression5). </Detailed reasoning> Answer: 1,14,4,5,2,4,7,20,9,10,11,12,13,14,3,8,3,18,16,6,18,11,5	1,14,17,19,2,4,7,20,9,10,15,23,12,6,22,8,3,18,16,13,21,11,5
gpto1	Detailed Reasoning: We are given a solution with missing expressions, and our goal is to correctly fill in these missing parts using the provided list of expressions. We'll proceed step by step, matching each missing part with the appropriate expression. 1. First Paragraph: - The solution starts with: > We can start by verifying that \( n=3 \) and <missing 4>* work by listing out the factors of \( 3! \) and \( 4! \).* - <missing 4> should be \( n=4 \) since the factors of \( 3! \) and \( 4! \) are being listed as examples that satisfy the condition. - Expression 5 is \( n=4 \). - So, <missing 4> = 5. - Next: > We can also see that \( n=5 \) does not work because the terms <missing 5>, and \( 24 \) are consecutive factors of \( 5! \). - The terms causing the violation are \( 15 \) and \( 20 \), with \( 24 \). - Expression 2 is "15, 20". - So, <missing 5> = 2. - Then: > Also, <missing 6>* does not work because the terms <missing 7>, and \( 9 \) appear consecutively in the factors of <missing 8>.* - <missing 6> should be \( n=6 \). - Expression 4 is \( n=6 \). - So, <missing 6> = 4. - <missing 7> are the terms \( 6, 8 \). - Expression 7 is "6, 8". - So, <missing 7> = 7. - <missing 8> refers to \( 6! \). - Expression 20 is \( 6! \). - So, <missing 8> = 20. 2. Second Paragraph: - The solution continues: > Note that if we have a prime number <missing 9>* and an integer <missing 10> such that both \( k \) and <missing 11> are factors of \( n! \), then the condition cannot be satisfied.* - <missing 9> is " \( p > n \) ". - Expression 9 is \( p > n \). - So, <missing 9> = 9. - <missing 10> is " \( k > p \) ". - Expression 10 is \( k > p \). - So, <missing 10> = 10. - <missing 11> is " \( k + 1 \) ". - Expression 15 is \( k + 1 \). - So, <missing 11> = 15. 3. Third Paragraph (Odd \( n \)): - Next, the solution analyzes odd \( n \): > If <missing 12>* is odd, then \( \(2\)\left(\frac{n - 1}{2}\right)(n - 1) = n^2 - 2n + 1 \) is a factor of \( n! \). Also, <missing 13> is a factor of <missing 14>.* - <missing 12> is "\( n \geq 7 \)". - Expression 3 is \( n \geq 7 \). - So, <missing 12> = 3. - <missing 13> refers to \( (n - 2)(n) = n^2 - 2n \). - Expression 12 is \( (n - 2)(n) = n^2 - 2n \). - So, <missing 13> = 12. - <missing 14> is \( n! \). - Expression 13 is \( n! \). - So, <missing 14> = 13. - Then: > Since <missing 15>* for all \( n \geq 7 \), we can use Bertrand's Postulate to show that there is at least one prime number \( p \) such that <missing 16>.* - <missing 15> is \( 2n < n^2 - 2n \). - Expression 22 is \( 2n < n^2 - 2n \). - So, <missing 15> = 22. - <missing 16> is \( n < p < n^2 - 2n \). - Expression 8 is \( n < p < n^2 - 2n \). - So, <missing 16> = 8. - Finally: > Since we have two consecutive factors of \( n! \) and a prime number between the smaller of these factors and \( n \), the condition will not be satisfied for all odd <missing 17>. - <missing 17> is "\( n \geq 7 \)". - Expression 23 is \( n \geq 7 \). - So, <missing 17> = 23. 4. Fourth Paragraph (Even \( n \)): - The solution now considers even \( n \): > If <missing 18>* is even, then <missing 19> is a factor of \( n! \). Also, \( (n - 3)(n - 1) = n^2 - 4n + 3 \) is a factor of \( n! \). Since \( 2n < n^2 - 4n + 3 \) for all \( n \geq 8 \), we can use Bertrand's Postulate again to show that there is at least one prime number \( p \) such that \( n < p < n^2 - 4n + 3 \).* - <missing 18> is "\( n \geq 8 \)". - Expression 18 is \( n \geq 8 \). - So, <missing 18> = 18. - <missing 19> is \( (n - 2)(n) = n^2 - 2n \). - Expression 12 is \( (n - 2)(n) = n^2 - 2n \). - So, <missing 19> = 12. - Then: > Since we have two consecutive factors of <missing 20>* and a prime number between the smaller of these factors and \( n \), the condition will not be satisfied for all even <missing 21>.* - <missing 20> is \( n! \). - Expression 6 is \( n! \). - So, <missing 20> = 6. - <missing 21> is "\( n \geq 8 \)". - Expression 21 is \( n \geq 8 \). - So, <missing 21> = 21. 5. Conclusion: - Finally: > Therefore, the only numbers that work are <missing 22>* and <missing 23>.* - <missing 22> is \( n=3 \). - Expression 11 is \( n=3 \). - So, <missing 22> = 11. - <missing 23> is \( n=4 \). - Expression 5 is \( n=4 \). - So, <missing 23> = 5. Answer: 5,2,4,20,1,2,4,7,20,9,10,15,3,12,13,22,8,23,18,12,6,21,11,5 --- Note: The numbers correspond to the expression identifiers that fill the missing parts in order. Here's the list of expressions used: - 5: \( n = 4 \) - 2: \( 15, 20 \) - 4: \( n = 6 \) - 20: \( 6! \) - 1: \( n = 5 \) - 7: \( 6, 8 \) - 9: \( p > n \) - 10: \( k > p \) - 15: \( k + 1 \) - 3: \( n \geq 7 \) - 12: \( (n - 2)(n) = n^2 - 2n \) - 13: \( n! \) - 22: \( 2n < n^2 - 2n \) - 8: \( n < p < n^2 - 2n \) - 23: \( n \geq 7 \) - 18: \( n \geq 8 \) - 6: \( n! \) - 21: \( n \geq 8 \) - 11: \( n = 3 \)	1,14,17,19,2,4,7,20,9,10,15,23,12,6,22,8,3,18,16,13,21,11,5
gpt-4o-2024-08-06	To solve this problem, we need to match the missing parts in the solution with the correct expressions. Let's go through the solution step by step and fill in the missing parts. 1. First Missing Part: - The solution mentions that $n=3$ and $n=4$ work, but $n=5$ does not work. The first missing part is likely referring to $n=5$. - Match: <missing 1> = <expression 1> (n=5) 2. Second Missing Part: - The second missing part is referring to the factorial of the number that does not work, which is $5!$. - Match: <missing 2> = <expression 14> (5!) 3. Third Missing Part: - The third missing part is about $n=6$ not working, as it mentions the terms $6, 8$, and $9$. - Match: <missing 3> = <expression 4> (n=6) 4. Fourth Missing Part: - The fourth missing part is about the numbers that work, which are $n=3$ and $n=4$. - Match: <missing 4> = <expression 5> (n=4) 5. Fifth Missing Part: - The fifth missing part is about the terms $15, 20$, and $24$ for $n=5$. - Match: <missing 5> = <expression 2> (15, 20) 6. Sixth Missing Part: - The sixth missing part is about $n=6$ not working. - Match: <missing 6> = <expression 17> (n=6) 7. Seventh Missing Part: - The seventh missing part is about the terms $6, 8$, and $9$ for $n=6$. - Match: <missing 7> = <expression 7> (6, 8) 8. Eighth Missing Part: - The eighth missing part is about the factorial of $n=6$, which is $6!$. - Match: <missing 8> = <expression 20> (6!) 9. Ninth Missing Part: - The ninth missing part is about a prime number greater than $n$. - Match: <missing 9> = <expression 9> (p>n) 10. Tenth Missing Part: - The tenth missing part is about an integer $k$ such that $k>p$. - Match: <missing 10> = <expression 10> (k>p) 11. Eleventh Missing Part: - The eleventh missing part is about $k+1$ being a factor of $n!$. - Match: <missing 11> = <expression 15> (k+1) 12. Twelfth Missing Part: - The twelfth missing part is about $(n-2)(n)=n^2-2n$ being odd. - Match: <missing 12> = <expression 12> ((n-2)(n)=n^2-2n) 13. Thirteenth Missing Part: - The thirteenth missing part is about $n!$ being a factor. - Match: <missing 13> = <expression 13> (n!) 14. Fourteenth Missing Part: - The fourteenth missing part is about $5!$. - Match: <missing 14> = <expression 14> (5!) 15. Fifteenth Missing Part: - The fifteenth missing part is about $2n<n^2-2n$ for $n\geq7$. - Match: <missing 15> = <expression 22> (2n<n^2-2n) 16. Sixteenth Missing Part: - The sixteenth missing part is about $n<p<n^2-2n$. - Match: <missing 16> = <expression 8> (n<p<n^2-2n) 17. Seventeenth Missing Part: - The seventeenth missing part is about odd $n$. - Match: <missing 17> = <expression 3> (n\geq7) 18. Eighteenth Missing Part: - The eighteenth missing part is about even $n$. - Match: <missing 18> = <expression 18> (n\geq8) 19. Nineteenth Missing Part: - The nineteenth missing part is about $n=4$. - Match: <missing 19> = <expression 19> (n=4) 20. Twentieth Missing Part: - The twentieth missing part is about $6!$. - Match: <missing 20> = <expression 20> (6!) 21. Twenty-first Missing Part: - The twenty-first missing part is about even $n$. - Match: <missing 21> = <expression 21> (n\geq8) 22. Twenty-second Missing Part: - The twenty-second missing part is about the numbers that work, which are $n=3$ and $n=4$. - Match: <missing 22> = <expression 11> (n=3) 23. Twenty-third Missing Part: - The twenty-third missing part is about the numbers that work, which are $n=3$ and $n=4$. - Match: <missing 23> = <expression 5> (n=4) Now, let's compile the final answer: Answer: 1, 14, 4, 5, 2, 17, 7, 20, 9, 10, 15, 12, 13, 14, 22, 8, 3, 18, 19, 20, 21, 11, 5	1,14,17,19,2,4,7,20,9,10,15,23,12,6,22,8,3,18,16,13,21,11,5
DeepSeek-Coder-V2-Lite-Instruct	To solve the problem, we need to match the missing tags in the solution with the corresponding formulae. Let's go through the solution step by step and identify the missing parts. 1. Verification for \( n = 3 \) and \( n = 4 \): - The solution mentions that \( n = 3 \) and \( n = 4 \) work. - This corresponds to the expression identifiers: \( n = 3 \) and \( n = 4 \). 2. Verification for \( n = 5 \): - The solution states that \( n = 5 \) does not work. - The terms \( 15, 20, \) and \( 24 \) are consecutive factors of \( 5! \). - This corresponds to the expression identifiers: \( 15, 20 \) and \( 24 \). 3. Verification for \( n = 6 \): - The solution mentions that \( n = 6 \) does not work. - The terms \( 6, 8, \) and \( 9 \) appear consecutively in the factors of \( 6! \). - This corresponds to the expression identifiers: \( 6, 8 \) and \( 9 \). 4. General Case Analysis: - The solution discusses the case where \( n \) is odd and even separately. - For odd \( n \), it mentions that \( (2)(\frac{n-1}{2})(n-1) = n^2 - 2n + 1 \) is a factor of \( n! \). - This corresponds to the expression identifier: \( (2)(\frac{n-1}{2})(n-1) = n^2 - 2n + 1 \). 5. Prime Number Analysis: - The solution uses Bertrand's Postulate to show that there is at least one prime number \( p \) such that \( n < p < n^2 - 2n \). - This corresponds to the expression identifier: \( n < p < n^2 - 2n \). 6. Conclusion: - The solution concludes that the only numbers that work are \( n = 2 \) and \( n = 3 \). - This corresponds to the expression identifiers: \( n = 2 \) and \( n = 3 \). Based on the above reasoning, the expression identifiers that fill the missing parts are: - \( n = 3 \) and \( n = 4 \) - \( 15, 20 \) and \( 24 \) - \( n \geq 7 \) - \( n = 6 \) - \( n = 4 \) - \( n! \) - \( 6, 8 \) and \( 9 \) - \( n < p < n^2 - 2n \) - \( p > n \) - \( k > p \) - \( n = 3 \) - \( (n-2)(n) = n^2 - 2n \) - \( n! \) - \( 5! \) - \( k + 1 \) - \( (2)(\frac{n-2}{2})(n-2) = n^2 - 4n + 4 \) - \( n = 6 \) - \( n \geq 8 \) - \( n = 4 \) - \( 6! \) - \( n \geq 8 \) - \( 2n < n^2 - 2n \) - \( n \geq 7 \) Answer: 5, 22, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23	1,14,17,19,2,4,7,20,9,10,15,23,12,6,22,8,3,18,16,13,21,11,5
Meta-Llama-3.1-405B-Instruct-Turbo	To solve this problem, we need to carefully read the solution and match the missing tags with the given formulae. The first missing tag is <missing 1>, which is described as a value of n that does not work. The corresponding formula is <expression 1>, which is n=5. The second missing tag is <missing 2>, which is described as the factorial of a number. The corresponding formula is <expression 6>, which is n!. The third missing tag is <missing 3>, which is described as a value of n that does not work. The corresponding formula is <expression 3>, which is n≥7. The fourth missing tag is <missing 4>, which is described as a value of n that works. The corresponding formula is <expression 5>, which is n=4. The fifth missing tag is <missing 5>, which is described as a term in the factors of 5!. The corresponding formula is <expression 2>, which is 15, 20. The sixth missing tag is <missing 6>, which is described as a value of n that does not work. The corresponding formula is <expression 4>, which is n=6. The seventh missing tag is <missing 7>, which is described as a term in the factors of 6!. The corresponding formula is <expression 7>, which is 6, 8. The eighth missing tag is <missing 8>, which is described as the factorial of a number. The corresponding formula is <expression 20>, which is 6!. The ninth missing tag is <missing 9>, which is described as a prime number. The corresponding formula is <expression 9>, which is p>n. The tenth missing tag is <missing 10>, which is described as an integer. The corresponding formula is <expression 10>, which is k>p. The eleventh missing tag is <missing 11>, which is described as a factor of n!. The corresponding formula is <expression 11>, which is n=3. The twelfth missing tag is <missing 12>, which is described as an odd number. The corresponding formula is <expression 12>, which is (n-2)(n)=n^2-2n. The thirteenth missing tag is <missing 13>, which is described as a factor of n!. The corresponding formula is <expression 13>, which is n!. The fourteenth missing tag is <missing 14>, which is described as the factorial of a number. The corresponding formula is <expression 14>, which is 5!. The fifteenth missing tag is <missing 15>, which is described as an inequality. The corresponding formula is <expression 22>, which is 2n<n^2-2n. The sixteenth missing tag is <missing 16>, which is described as an inequality. The corresponding formula is <expression 8>, which is n<p<n^2-2n. The seventeenth missing tag is <missing 17>, which is described as an odd number. The corresponding formula is <expression 17>, which is n=6, but this is incorrect, so we will use <expression 3>, which is n≥7. The eighteenth missing tag is <missing 18>, which is described as an even number. The corresponding formula is <expression 18>, which is n≥8. The nineteenth missing tag is <missing 19>, which is described as a factor of n!. The corresponding formula is <expression 19>, which is n=4, but this is incorrect, so we will use <expression 16>, which is (2)(n-2)/2)(n-2)=n^2-4n+4. The twentieth missing tag is <missing 20>, which is described as the factorial of a number. The corresponding formula is <expression 6>, which is n!. The twenty-first missing tag is <missing 21>, which is described as an even number. The corresponding formula is <expression 21>, which is n≥8. The twenty-second missing tag is <missing 22>, which is described as a number that works. The corresponding formula is <expression 11>, which is n=3. The twenty-third missing tag is <missing 23>, which is described as a number that works. The corresponding formula is <expression 5>, which is n=4. Answer: 1, 6, 3, 5, 2, 4, 7, 20, 9, 10, 11, 12, 13, 14, 22, 8, 3, 18, 16, 6, 21, 11, 5	1,14,17,19,2,4,7,20,9,10,15,23,12,6,22,8,3,18,16,13,21,11,5
Meta-Llama-3.1-70B-Instruct-Turbo	To solve this problem, we need to carefully match the missing tags in the solution with the given formulae. Let's go through the solution step by step and identify the corresponding formulae. 1. The first missing tag is <missing 1>, which corresponds to the expression "n=5". This is because the solution states that "n=5 does not work". Therefore, <missing 1> = <expression 1>. 2. The next missing tag is <missing 2>, which corresponds to the expression "15, 20". This is because the solution states that "the terms 15, 20, and 24 are consecutive factors of <missing 2>". Therefore, <missing 2> = <expression 2>. 3. The next missing tag is <missing 3>, which corresponds to the expression "n=6". This is because the solution states that "<missing 3> does not work because the terms 6, 8, and 9 appear consecutively in the factors of 6!". Therefore, <missing 3> = <expression 4>. 4. The next missing tag is <missing 4>, which corresponds to the expression "n=4". This is because the solution states that "n=3 and <missing 4> work by listing out the factors of 3! and 4!". Therefore, <missing 4> = <expression 5>. 5. The next missing tag is <missing 5>, which corresponds to the expression "6, 8". This is because the solution states that "n=5 does not work because the terms <missing 5>, and 24 are consecutive factors of 5!". Therefore, <missing 5> = <expression 7>. 6. The next missing tag is <missing 6>, which corresponds to the expression "n=6". This is because the solution states that "<missing 6> does not work because the terms <missing 7>, and 9 appear consecutively in the factors of <missing 8>". Therefore, <missing 6> = <expression 4>. 7. The next missing tag is <missing 7>, which corresponds to the expression "6, 8". This is because the solution states that "<missing 6> does not work because the terms <missing 7>, and 9 appear consecutively in the factors of <missing 8>". Therefore, <missing 7> = <expression 7>. 8. The next missing tag is <missing 8>, which corresponds to the expression "6!". This is because the solution states that "<missing 6> does not work because the terms <missing 7>, and 9 appear consecutively in the factors of <missing 8>". Therefore, <missing 8> = <expression 20>. 9. The next missing tag is <missing 9>, which corresponds to the expression "p>n". This is because the solution states that "if we have a prime number <missing 9> and an integer <missing 10> such that both k and <missing 11> are factors of n!, then the condition cannot be satisfied". Therefore, <missing 9> = <expression 9>. 10. The next missing tag is <missing 10>, which corresponds to the expression "k>p". This is because the solution states that "if we have a prime number <missing 9> and an integer <missing 10> such that both k and <missing 11> are factors of n!, then the condition cannot be satisfied". Therefore, <missing 10> = <expression 10>. 11. The next missing tag is <missing 11>, which corresponds to the expression "n=3". This is because the solution states that "if we have a prime number <missing 9> and an integer <missing 10> such that both k and <missing 11> are factors of n!, then the condition cannot be satisfied". Therefore, <missing 11> = <expression 11>. 12. The next missing tag is <missing 12>, which corresponds to the expression "(n-2)(n)=n^2-2n". This is because the solution states that "if <missing 12> is odd, then (2)(\frac{n-1}{2})(n-1)=n^2-2n+1 is a factor of n!". Therefore, <missing 12> = <expression 12>. 13. The next missing tag is <missing 13>, which corresponds to the expression "n!". This is because the solution states that "<missing 13> is a factor of <missing 14>". Therefore, <missing 13> = <expression 13>. 14. The next missing tag is <missing 14>, which corresponds to the expression "5!". This is because the solution states that "<missing 13> is a factor of <missing 14>". Therefore, <missing 14> = <expression 14>. 15. The next missing tag is <missing 15>, which corresponds to the expression "k+1". This is because the solution states that "since <missing 15> for all n\geq7, we can use Bertrand's Postulate to show that there is at least one prime number p such that <missing 16>". Therefore, <missing 15> = <expression 15>. 16. The next missing tag is <missing 16>, which corresponds to the expression "(2)(\frac{n-2}{2})(n-2)=n^2-4n+4". This is because the solution states that "since <missing 15> for all n\geq7, we can use Bertrand's Postulate to show that there is at least one prime number p such that <missing 16>". Therefore, <missing 16> = <expression 16>. 17. The next missing tag is <missing 17>, which corresponds to the expression "n=6". This is because the solution states that "the condition will not be satisfied for all odd <missing 17>". Therefore, <missing 17> = <expression 17>. 18. The next missing tag is <missing 18>, which corresponds to the expression "n\geq8". This is because the solution states that "if <missing 18> is even, then <missing 19> is a factor of n!". Therefore, <missing 18> = <expression 18>. 19. The next missing tag is <missing 19>, which corresponds to the expression "n=4". This is because the solution states that "if <missing 18> is even, then <missing 19> is a factor of n!". Therefore, <missing 19> = <expression 19>. 20. The next missing tag is <missing 20>, which corresponds to the expression "6!". This is because the solution states that "since 2n<n^2-4n+3 for all n\geq8, we can use Bertrand's Postulate again to show that there is at least one prime number p such that n<p<n^2-4n+3. Since we have two consecutive factors of <missing 20> and a prime number between the smaller of these factors and n, the condition will not be satisfied for all even <missing 21>". Therefore, <missing 20> = <expression 20>. 21. The next missing tag is <missing 21>, which corresponds to the expression "n\geq8". This is because the solution states that "since 2n<n^2-4n+3 for all n\geq8, we can use Bertrand's Postulate again to show that there is at least one prime number p such that n<p<n^2-4n+3. Since we have two consecutive factors of <missing 20> and a prime number between the smaller of these factors and n, the condition will not be satisfied for all even <missing 21>". Therefore, <missing 21> = <expression 21>. 22. The next missing tag is <missing 22>, which corresponds to the expression "2n<n^2-2n". This is because the solution states that "therefore, the only numbers that work are <missing 22> and <missing 23>". Therefore, <missing 22> = <expression 22>. 23. The next missing tag is <missing 23>, which corresponds to the expression "n\geq7". This is because the solution states that "therefore, the only numbers that work are <missing 22> and <missing 23>". Therefore, <missing 23> = <expression 23>. Answer: 1, 2, 4, 5, 7, 4, 20, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23	1,14,17,19,2,4,7,20,9,10,15,23,12,6,22,8,3,18,16,13,21,11,5
Qwen2-72B-Instruct	The solution mentions that $n=5$ does not work because the terms $15, 20$ , and $24$ are consecutive factors of $5!$. This corresponds to expressions 1, 2, and 14. Then it says that $n=6$ does not work because the terms $6, 8$ , and $9$ appear consecutively in the factors of $6!$. This corresponds to expressions 4, 7, and 20. It then goes on to say that if $n$ is odd and greater than or equal to 7, then the condition cannot be satisfied due to Bertrand's Postulate. This corresponds to expression 3. Similarly, if $n$ is even and greater than or equal to 8, then the condition cannot be satisfied due to Bertrand's Postulate. This corresponds to expression 18. The solution then concludes that the only numbers that work are $n=3$ and $n=4$. This corresponds to expressions 11 and 5. Answer: 1, 2, 3, 4, 5, 7, 20, 3, 12, 13, 14, 15, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 11, 5	1,14,17,19,2,4,7,20,9,10,15,23,12,6,22,8,3,18,16,13,21,11,5
baichuan4	To solve this problem, we need to match the missing tags in the solution with the provided formulae. Let's go through the solution step by step and identify which formulae fit into the missing tags. 1. The first missing tag is <missing 1>. The context suggests that it should be a value of $n$ that does not work. The only formula that fits this description is <expression 1> which is $n=5$. 2. The second missing tag is <missing 2>. This should be a factorial that has consecutive factors mentioned. The only factorial mentioned in the formulae list is <expression 6> which is $n!$. However, we need to specify which factorial, so we'll keep this in mind and revisit it later. 3. The third missing tag is <missing 3>. This should be a value of $n$ that does not work. The only formula that fits this description and hasn't been used yet is <expression 4> which is $n=6$. 4. The fourth missing tag is <missing 4>. This should be a value of $n$ that works, as it's mentioned along with $n=3$. The only formula that fits this description and hasn't been used yet is <expression 5> which is $n=4$. 5. The fifth missing tag is <missing 5>. This should be a sequence of consecutive factors. The only sequence that fits this description is <expression 2> which is $15, 20$. 6. The sixth missing tag is <missing 6>. This should be a value of $n$ that does not work. The only formula that fits this description and hasn't been used yet is <expression 17> which is $n=6$. 7. The seventh missing tag is <missing 7>. This should be a sequence of consecutive factors. The only sequence that fits this description and hasn't been used yet is <expression 7> which is $6, 8$. 8. The eighth missing tag is <missing 8>. This should be a factorial that has the sequence of factors mentioned in <missing 7>. The only factorial mentioned in the formulae list that hasn't been used yet is <expression 20> which is $6!$. 9. The ninth missing tag is <missing 9>. This should be a prime number condition. The only formula that fits this description is <expression 9> which is $p>n$. 10. The tenth missing tag is <missing 10>. This should be an integer condition related to the prime number mentioned in <missing 9>. The only formula that fits this description is <expression 10> which is $k>p$. 11. The eleventh missing tag is <missing 11>. This should be a value of $n$ that works. The only formula that fits this description and hasn't been used yet is <expression 11> which is $n=3$. 12. The twelfth missing tag is <missing 12>. This should be an expression related to odd values of $n$. The only formula that fits this description is <expression 12> which is $(n-2)(n)=n^2-2n$. 13. The thirteenth missing tag is <missing 13>. This should be a factorial. The only factorial mentioned in the formulae list that hasn't been used yet is <expression 13> which is $n!$. 14. The fourteenth missing tag is <missing 14>. This should be a specific factorial related to the one mentioned in <missing 13>. The only factorial mentioned in the formulae list that hasn't been used yet is <expression 14> which is $5!$. 15. The fifteenth missing tag is <missing 15>. This should be a condition related to consecutive factors. The only formula that fits this description is <expression 15> which is $k+1$. 16. The sixteenth missing tag is <missing 16>. This should be an expression related to even values of $n$. The only formula that fits this description is <expression 16> which is $(2)(\frac{n-2}{2})(n-2)=n^2-4n+4$. 17. The seventeenth missing tag is <missing 17>. This should be a condition related to odd values of $n$. The only formula that fits this description and hasn't been used yet is <expression 23> which is $n\geq7$. 18. The eighteenth missing tag is <missing 18>. This should be a condition related to even values of $n$. The only formula that fits this description is <expression 18> which is $n\geq8$. 19. The nineteenth missing tag is <missing 19>. This should be a specific value of $n$ related to even values. The only formula that fits this description and hasn't been used yet is <expression 19> which is $n=4$. 20. The twentieth missing tag is <missing 20>. This should be a factorial related to even values of $n$. The only factorial mentioned in the formulae list that hasn't been used yet is <expression 20> which is $6!$. 21. The twenty-first missing tag is <missing 21>. This should be a condition related to even values of $n$. The only formula that fits this description and hasn't been used yet is <expression 21> which is $n\geq8$. 22. The twenty-second missing tag is <missing 22>. This should be the final conclusion for values of $n$ that work. The only formula that fits this description and hasn't been used yet is <expression 11> which is $n=3$. 23. The twenty-third missing tag is <missing 23>. This should be the final conclusion for values of $n$ that work. The only formula that fits this description and hasn't been used yet is <expression 5> which is $n=4$. Therefore, the answer is: <Detailed reasoning> Answer: 1, 6, 4, 5, 2, 17, 7, 20, 9, 10, 11, 12, 13, 14, 15, 16, 23, 18, 19, 20, 21, 11, 5	1,14,17,19,2,4,7,20,9,10,15,23,12,6,22,8,3,18,16,13,21,11,5
doubao	<Detailed reasoning>: In the solution, <missing 1> is related to a case that doesn't work, and expression 1 is "n=5" which is the value that doesn't work, so 1 fills <missing 1>. <missing 2> is related to the factors of 5!, and expression 14 is "5!" so 14 fills <missing 2>. <missing 3> is related to a case that doesn't work, and expression 4 is "n=6" which is the value that doesn't work, so 4 fills <missing 3>. <missing 4> is for the other value that works initially, and expression 5 is "n=4" so 5 fills <missing 4>. <missing 5> is related to the consecutive factors of 5!, and expression 2 is "15, 20" so 2 fills <missing 5>. <missing 6> is related to the factors of 6!, and expression 6 is "n!" so 6 fills <missing 6>. <missing 7> is related to the consecutive factors of 6!, and expression 7 is "6, 8" so 7 fills <missing 7>. <missing 8> is related to the factors of a specific value, and expression 20 is "6!" so 20 fills <missing 8>. <missing 9> is related to a prime number condition, and expression 9 is "p>n" so 9 fills <missing 9>. <missing 10> is related to an integer condition with a prime, and expression 10 is "k>p" so 10 fills <missing 10>. <missing 11> is the other value that works initially, and expression 11 is "n=3" so 11 fills <missing 11>. <missing 12> is for an odd case, and expression 12 is "(n-2)(n)=n^2-2n" so 12 fills <missing 12>. <missing 13> is related to a factor, and expression 13 is "n!" so 13 fills <missing 13>. <missing 15> is related to a comparison, and expression 15 is "k+1" so 15 fills <missing 15>. <missing 16> is for a specific calculation, and expression 16 is "(2)(\frac{n-2}{2})(n-2)=n^2-4n+4" so 16 fills <missing 16>. <missing 17	1,14,17,19,2,4,7,20,9,10,15,23,12,6,22,8,3,18,16,13,21,11,5
gpto1	Detailed Reasoning: We are given a solution with missing expressions, and our goal is to correctly fill in these missing parts using the provided list of expressions. We'll proceed step by step, matching each missing part with the appropriate expression. 1. First Paragraph: - The solution starts with: > We can start by verifying that \( n=3 \) and <missing 4>* work by listing out the factors of \( 3! \) and \( 4! \).* - <missing 4> should be \( n=4 \) since the factors of \( 3! \) and \( 4! \) are being listed as examples that satisfy the condition. - Expression 5 is \( n=4 \). - So, <missing 4> = 5. - Next: > We can also see that \( n=5 \) does not work because the terms <missing 5>, and \( 24 \) are consecutive factors of \( 5! \). - The terms causing the violation are \( 15 \) and \( 20 \), with \( 24 \). - Expression 2 is "15, 20". - So, <missing 5> = 2. - Then: > Also, <missing 6>* does not work because the terms <missing 7>, and \( 9 \) appear consecutively in the factors of <missing 8>.* - <missing 6> should be \( n=6 \). - Expression 4 is \( n=6 \). - So, <missing 6> = 4. - <missing 7> are the terms \( 6, 8 \). - Expression 7 is "6, 8". - So, <missing 7> = 7. - <missing 8> refers to \( 6! \). - Expression 20 is \( 6! \). - So, <missing 8> = 20. 2. Second Paragraph: - The solution continues: > Note that if we have a prime number <missing 9>* and an integer <missing 10> such that both \( k \) and <missing 11> are factors of \( n! \), then the condition cannot be satisfied.* - <missing 9> is " \( p > n \) ". - Expression 9 is \( p > n \). - So, <missing 9> = 9. - <missing 10> is " \( k > p \) ". - Expression 10 is \( k > p \). - So, <missing 10> = 10. - <missing 11> is " \( k + 1 \) ". - Expression 15 is \( k + 1 \). - So, <missing 11> = 15. 3. Third Paragraph (Odd \( n \)): - Next, the solution analyzes odd \( n \): > If <missing 12>* is odd, then \( \(2\)\left(\frac{n - 1}{2}\right)(n - 1) = n^2 - 2n + 1 \) is a factor of \( n! \). Also, <missing 13> is a factor of <missing 14>.* - <missing 12> is "\( n \geq 7 \)". - Expression 3 is \( n \geq 7 \). - So, <missing 12> = 3. - <missing 13> refers to \( (n - 2)(n) = n^2 - 2n \). - Expression 12 is \( (n - 2)(n) = n^2 - 2n \). - So, <missing 13> = 12. - <missing 14> is \( n! \). - Expression 13 is \( n! \). - So, <missing 14> = 13. - Then: > Since <missing 15>* for all \( n \geq 7 \), we can use Bertrand's Postulate to show that there is at least one prime number \( p \) such that <missing 16>.* - <missing 15> is \( 2n < n^2 - 2n \). - Expression 22 is \( 2n < n^2 - 2n \). - So, <missing 15> = 22. - <missing 16> is \( n < p < n^2 - 2n \). - Expression 8 is \( n < p < n^2 - 2n \). - So, <missing 16> = 8. - Finally: > Since we have two consecutive factors of \( n! \) and a prime number between the smaller of these factors and \( n \), the condition will not be satisfied for all odd <missing 17>. - <missing 17> is "\( n \geq 7 \)". - Expression 23 is \( n \geq 7 \). - So, <missing 17> = 23. 4. Fourth Paragraph (Even \( n \)): - The solution now considers even \( n \): > If <missing 18>* is even, then <missing 19> is a factor of \( n! \). Also, \( (n - 3)(n - 1) = n^2 - 4n + 3 \) is a factor of \( n! \). Since \( 2n < n^2 - 4n + 3 \) for all \( n \geq 8 \), we can use Bertrand's Postulate again to show that there is at least one prime number \( p \) such that \( n < p < n^2 - 4n + 3 \).* - <missing 18> is "\( n \geq 8 \)". - Expression 18 is \( n \geq 8 \). - So, <missing 18> = 18. - <missing 19> is \( (n - 2)(n) = n^2 - 2n \). - Expression 12 is \( (n - 2)(n) = n^2 - 2n \). - So, <missing 19> = 12. - Then: > Since we have two consecutive factors of <missing 20>* and a prime number between the smaller of these factors and \( n \), the condition will not be satisfied for all even <missing 21>.* - <missing 20> is \( n! \). - Expression 6 is \( n! \). - So, <missing 20> = 6. - <missing 21> is "\( n \geq 8 \)". - Expression 21 is \( n \geq 8 \). - So, <missing 21> = 21. 5. Conclusion: - Finally: > Therefore, the only numbers that work are <missing 22>* and <missing 23>.* - <missing 22> is \( n=3 \). - Expression 11 is \( n=3 \). - So, <missing 22> = 11. - <missing 23> is \( n=4 \). - Expression 5 is \( n=4 \). - So, <missing 23> = 5. Answer: 5,2,4,20,1,2,4,7,20,9,10,15,3,12,13,22,8,23,18,12,6,21,11,5 --- Note: The numbers correspond to the expression identifiers that fill the missing parts in order. Here's the list of expressions used: - 5: \( n = 4 \) - 2: \( 15, 20 \) - 4: \( n = 6 \) - 20: \( 6! \) - 1: \( n = 5 \) - 7: \( 6, 8 \) - 9: \( p > n \) - 10: \( k > p \) - 15: \( k + 1 \) - 3: \( n \geq 7 \) - 12: \( (n - 2)(n) = n^2 - 2n \) - 13: \( n! \) - 22: \( 2n < n^2 - 2n \) - 8: \( n < p < n^2 - 2n \) - 23: \( n \geq 7 \) - 18: \( n \geq 8 \) - 6: \( n! \) - 21: \( n \geq 8 \) - 11: \( n = 3 \)	1,14,17,19,2,4,7,20,9,10,15,23,12,6,22,8,3,18,16,13,21,11,5
kimi	<Detailed reasoning> 1. The solution mentions that $n=3$ and $n=4$ work, so <missing 5> should be replaced with <expression 1> n=5. 2. The solution also mentions that $n=5$ does not work because the terms 15, 20 are consecutive factors of $5!$. So <missing 2> should be replaced with <expression 2> 15, 20. 3. The solution states that <missing 3> does not work because the terms 6, 8 appear consecutively in the factors of $6!$. So <missing 3> should be replaced with <expression 4> n=6. 4. The solution mentions that if we have a prime number <missing 9> and an integer <missing 10> such that both $k$ and <missing 11> are factors of $n!$, then the condition cannot be satisfied. Since <missing 9> is a prime number and <missing 10> is an integer, we can assume that <missing 9> is replaced with <expression 9> p>n and <missing 10> is replaced with <expression 10> k>p. 5. The solution mentions that if <missing 12> is odd, then $(2)(\frac{n-1}{2})(n-1)=n^2-2n+1$ is a factor of $n!$. So <missing 12> should be replaced with <expression 12> (n-2)(n)=n^2-2n. 6. The solution mentions that <missing 13> is a factor of <missing 14>. Since <missing 14> is $5!$, we can assume that <missing 13> is replaced with <expression 13> n!. 7. The solution states that since <missing 15> for all $n\geq7$, we can use Bertrand's Postulate to show that there is at least one prime number $p$ such that <missing 16>. Since <missing 15> is $k+1$, we can assume that <missing 15> is replaced with <expression 15> k+1. 8. The solution mentions that if <missing 18> is even, then <missing 19> is a factor of $n!$. Since <missing 18> is even and <missing 19> is a factor of $n!$, we can assume that <missing 18> is replaced with <expression 18> n\geq8 and <missing 19> is replaced with <expression 19> n=4. 9. The solution mentions that since $2n<n^2-4n+3$ for all $n\geq8$, we can use Bertrand's Postulate again to show that there is at least one prime number $p$ such that $n<p<n^2-4n+3$. So <missing 20> should be replaced with <expression 20> 6!. 10. The solution states that the only numbers that work are <missing 22> and <missing 23>. Since <missing 22> is $2n<n^2-2n$ and <missing 23> is $n\geq7$, we can assume that <missing 22> is replaced with <expression 22> 2n<n^2-2n and <missing 23> is replaced with <expression 23> n\geq7. Answer: 1,2,4,6,9,10,12,13,15,18,19,20,22,23	1,14,17,19,2,4,7,20,9,10,15,23,12,6,22,8,3,18,16,13,21,11,5
minimax	To solve this problem, we need to match the missing parts in the solution with the corresponding expressions provided. Let's go through the solution step by step and match the expressions to the missing tags. 1. The solution starts by verifying that $n=3$ and $n=4$ work. This is straightforward and doesn't require any formulae from the list. 2. The solution then states that $n=5$ does not work because the terms <missing 5> and $24$ are consecutive factors of $5!$. Since $24$ is a factor of $5!$, we need to find a consecutive factor that, when subtracted from $24$, gives a result that would violate the condition. The only consecutive factor of $5!$ that would do this is $20$, because $24 - 20 = 4$ and $20 - 15 = 5$, which does not satisfy the condition $d_2-d_1 \leq d_3-d_2$. Therefore, <missing 5> corresponds to $15$. 3. The solution also mentions that <missing 6> does not work because the terms <missing 7> and $9$ appear	1,14,17,19,2,4,7,20,9,10,15,23,12,6,22,8,3,18,16,13,21,11,5
wenxinyiyan	First, let's analyze the given solution step by step. 1. The solution starts by verifying that $n=3$ and $n=4$ work. This gives us the first hint that <missing 1> could be $n=5$ since it's the next integer to be checked. 2. Next, the solution says "<missing 1> does not work because the terms $15, 20$, and $24$ are consecutive factors of <missing 2>". This confirms that <missing 1> is indeed $n=5$ and <missing 2> is $5!$ because $15, 20, 24$ are consecutive factors of $5!$. 3. The solution then says "<missing 3> does not work because the terms $6, 8$, and $9$ appear consecutively in the factors of $6!$". This tells us that <missing 3> is $n=6$. 4. "<missing 4> work by listing out the factors of $3!$ and $4!$". This means <missing 4> is $n=4$ since it's the only other value besides $n=3$ that was verified to work. 5. "the terms <missing 5>, and $24$ are consecutive factors of $5!$". This tells us that <missing 5> is $15, 20$ since these are the terms mentioned earlier that are consecutive factors of $5!$. 6. "<missing 6> does not work because the terms <missing 7>, and $9$ appear consecutively in the factors of <missing 8>". This means <missing 6> is $n=6$ (again) and <missing 7> is $6, 8$ since these are the terms mentioned. <missing 8> must be $6!$ since it's talking about the factors of $6!$. 7. "If <missing 9> and <missing 10> such that both $k$ and <missing 11> are factors of $n!$". This part is tricky because it's not clear what $k$ is, but since <missing 11> is mentioned in the context of factors, it's likely that <missing 11> is a factor of $n!$ as well. Given the previous information, <missing 11> could be $n=3$ since $3!$ was one of the factorials that worked. 8. "If <missing 12> is odd, then $(2)(\frac{n-1}{2})(n-1)=n^2-2n+1$ is a factor of $n!$". This means <missing 12> is likely talking about $n$ being odd. 9. "<missing 13> is a factor of <missing 14>". Given the context, <missing 13> is likely $n!$ again and <missing 14> is $5!$ since it's talking about factors. 10. "Since <missing 15> for all $n\geq7$". This part is unclear without more context, but it seems to be leading into a discussion about Bertrand's Postulate, so <missing 15> could be related to the conditions needed for the postulate to apply. 11. "we can use Bertrand's Postulate to show that there is at least one prime number $p$ such that <missing 16>". This means <missing 16> is likely a condition related to Bertrand's Postulate, such as $n<p<n^2-2n$. 12. "the condition will not be satisfied for all odd <missing 17>". Given the context, <missing 17> is likely $n$ being odd again. 13. "If <missing 18> is even, then <missing 19> is a factor of $n!$". This means <missing 18> is likely talking about $n$ being even and <missing 19> is a factor of $n!$ when $n$ is even. Given the previous information, <missing 19> could be $(n-2)(n)=n^2-2n$ since it's a factor of $n!$ when $n$ is even. 14. "Also, $(n-3)(n-1)=n^2-4n+3$ is a factor of $n!$". This confirms that <missing 19> is indeed $(n-2)(n)=n^2-2n$ since it's listed as another factor. 15. "Since $2n<n^2-4n+	1,14,17,19,2,4,7,20,9,10,15,23,12,6,22,8,3,18,16,13,21,11,5
zhipu	### Detailed Reasoning 1. <missing 1>: The context suggests that we are looking for an integer \( n \) that does not satisfy the given property. The terms \( 15, 20 \), and \( 24 \) are consecutive factors of \( n! \). This implies \( n = 5 \) because these numbers are factors of \( 5! \). Thus, <missing 1> should be \( n = 5 \). - Match: <expression 1> 2. <missing 2>: This should be the factorial of the number mentioned in <missing 1>, which is \( 5! \). - Match: <expression 14> 3. <missing 3>: We need an integer \( n \) where \( 6, 8 \), and \( 9 \) are consecutive factors of \( n! \). This implies \( n = 6 \) because these numbers are factors of \( 6! \). - Match: <expression 4> 4. <missing 4>: This should be the next integer after \( n = 3 \) that satisfies the property, which is \( n = 4 \). - Match: <expression 5> 5. <missing 5>: These are the terms that, along with \( 24 \), are consecutive factors of \( 5! \). The terms are \( 15 \) and \( 20 \). - Match: <expression 2> 6. <missing 6>: This should be the integer \( n \) where \( 6, 8 \), and \( 9 \) are consecutive factors of \( n! \), which is \( n = 6 \). - Match: <expression 4> 7. <missing 7>: These are the terms that, along with \( 9 \), are consecutive factors of \( 6! \). The terms are \( 6 \) and \( 8 \). - Match: <expression 7> 8. <missing 8>: This should be the factorial of the number mentioned in <missing 6>, which is \( 6! \). - Match: <expression 20> 9. <missing 9>: This is a prime number \( p \). - Match: <expression 9> 10. <missing 10>: This is an integer \( k \) such that \( k > p \). - Match: <expression 10> 11. <missing 11>: This should be \( k+1 \) since both \( k \) and \( k+1 \) are factors of \( n! \). - Match: <expression 15> 12. <missing 12>: This refers to \( n \) being odd. - Match: <expression 11> 13. <missing 13>: This should be \( n! \). - Match: <expression 13> 14. <missing 14>: This should be \( n! \). - Match: <expression 13> 15. <missing 15>: This is a condition for \( n \), which is \( n \geq 7 \). - Match: <expression 3> 16. <missing 16>: This is a condition involving a prime number \( p \) such that \( n < p < n^2 - 2n \). - Match: <expression 8> 17. <missing 17>: This refers to all odd \( n \geq 7 \). - Match: <expression 23> 18. <missing 18>: This refers to \( n \) being even. - Match: <expression 19> 19. <missing 19>: This should be \( n! \). - Match: <expression 13> 20. <missing 20>: This should be \( n! \). - Match: <expression 13> 21. <missing 21>: This refers to all even \( n \geq 8 \). - Match: <expression 21> 22. <missing 22>: This is one of the numbers that work, which is \( n = 3 \). - Match: <expression 11> 23. <missing 23>: This is the other number that works, which is \( n = 4 \). - Match: <expression 5> ### Answer 1, 14, 4, 5, 2, 4, 7, 20, 9, 10, 15, 11, 13, 13, 3, 8, 23, 19, 13, 13, 21, 11, 5	1,14,17,19,2,4,7,20,9,10,15,23,12,6,22,8,3,18,16,13,21,11,5
deepseek-v3	Error code: 400 - {'error': {'code': 'RequestTimeOut', 'param': None, 'message': 'Request timed out, please try again later.', 'type': 'RequestTimeOut'}, 'id': 'chatcmpl-fc67d279-e3d4-9a34-8f14-31acf5d763f5', 'request_id': 'fc67d279-e3d4-9a34-8f14-31acf5d763f5'}	1,14,17,19,2,4,7,20,9,10,15,23,12,6,22,8,3,18,16,13,21,11,5

模型

模型答案

正确答案

金声玉亮2.0

<Detailed reasoning>

To match the formulae to the missing tags in the solution, we need to carefully analyze the context in which each <missing X> appears and find the corresponding expression from the list. Here's the step-by-step reasoning:

1. **<missing 1>**: The solution mentions that a certain value of $n$ does not work due to specific terms being consecutive factors. The context suggests it's about a specific $n$ value, and the only expression directly mentioning a specific $n$ value in this context is <expression 1> $n=5$.
2. **<missing 2>**: This refers to the factors of a specific $n!$, and the context mentions $15, 20, 24$ as consecutive factors. The only expression listing these terms is <expression 2> $15, 20$.
3. **<missing 3>**: Similar to <missing 1>, this refers to a specific $n$ value that does not work. The context mentions $n=6$, which corresponds to <expression 4> $n=6$.
4. **<missing 4>**: This refers to a specific $n$ value that works, and the context mentions $n=4$, which corresponds to <expression 5> $n=4$.
5. **<missing 5>**: This refers to the terms $15, 20$ being consecutive factors of $5!$, which corresponds to <expression 2> $15, 20$.
6. **<missing 6>**: This refers to a specific $n$ value that does not work, and the context mentions $n=6$, which corresponds to <expression 4> $n=6$.
7. **<missing 7>**: This refers to the terms $6, 8$ being consecutive factors of $6!$, which corresponds to <expression 7> $6, 8$.
8. **<missing 8>**: This refers to the factors of $6!$, which corresponds to <expression 20> $6!$.
9. **<missing 9>**: This refers to a prime number $p$ greater than $n$, which corresponds to <expression 9> $p>n$.
10. **<missing 10>**: This refers to an integer $k$ such that $k>p$, which corresponds to <expression 10> $k>p$.
11. **<missing 11>**: This refers to $k+1$, which corresponds to <expression 15> $k+1$.
12. **<missing 12>**: This refers to an odd $n$, and the context suggests it's about a specific condition for odd $n$, which corresponds to <expression 3> $n\geq7$.
13. **<missing 13>**: This refers to a factor of $n!$, which corresponds to <expression 13> $n!$.
14. **<missing 14>**: This refers to a specific $n!$, and the context mentions $5!$, which corresponds to <expression 14> $5!$.
15. **<missing 15>**: This refers to a condition for all $n\geq7$, which corresponds to <expression 22> $2n<n^2-2n$.
16. **<missing 16>**: This refers to a prime number $p$ such that $n<p<n^2-2n$, which corresponds to <expression 8> $n<p<n^2-2n$.
17. **<missing 17>**: This refers to all odd $n\geq7$, which corresponds to <expression 3> $n\geq7$.
18. **<missing 18>**: This refers to an even $n$, and the context suggests it's about a specific condition for even $n$, which corresponds to <expression 18> $n\geq8$.
19. **<missing 19>**: This refers to a specific even $n$ value, and the context mentions $n=4$, which corresponds to <expression 5> $n=4$.
20. **<missing 20>**: This refers to the factors of $n!$, which corresponds to <expression 13> $n!$.
21. **<missing 21>**: This refers to all even $n\geq8$, which corresponds to <expression 21> $n\geq8$.
22. **<missing 22>**: This refers to a specific $n$ value that works, and the context mentions $n=3$, which corresponds to <expression 11> $n=3$.
23. **<missing 23>**: This refers to a specific $n$ value that works, and the context mentions $n=4$, which corresponds to <expression 5> $n=4$.

Answer: 1, 2, 4, 5, 2, 4, 7, 20, 9, 10, 15, 3, 13, 14, 22, 8, 3, 18, 5, 13, 21, 11, 5

1,14,17,19,2,4,7,20,9,10,15,23,12,6,22,8,3,18,16,13,21,11,5

deepseek-r1

<Detailed reasoning>
The solution checks small n values first. For missing1, the example with terms 15,20,24 in 5! implies n=5 (expression1). missing2 is 5! (expression14). For n=6 not working (missing3), use expression4. missing4 refers to n=4 (expression5). missing5 lists 15,20 (expression2). missing6 is n=6 (expression4), and missing7 is 6,8 (expression7). missing8 is 6! (expression20). For primes, missing9 is p>n (expression9), missing10 is k>p (expression10), and missing11 is n=3 (expression11). When n is odd, missing12 is the formula (expression12). missing13 is n! (expression13), missing14 is 5! (expression14). missing15 refers to n≥7 (expression3). Bertrand's interval is expression8. missing17 is n≥7 (expression3). For even n≥8 (expression18), missing19 is the formula (expression16). missing20 is n! (expression6), and missing21 is n≥8 (expression18). Finally, the valid n's are 3 (expression11) and 4 (expression5).
</Detailed reasoning>

Answer: 1,14,4,5,2,4,7,20,9,10,11,12,13,14,3,8,3,18,16,6,18,11,5

1,14,17,19,2,4,7,20,9,10,15,23,12,6,22,8,3,18,16,13,21,11,5

gpto1

**Detailed Reasoning:**

We are given a solution with missing expressions, and our goal is to correctly fill in these missing parts using the provided list of expressions. We'll proceed step by step, matching each missing part with the appropriate expression.

1. **First Paragraph:**

- The solution starts with:
> *We can start by verifying that $ n=3 $ and **<missing 4>** work by listing out the factors of $ 3! $ and $ 4! $.*

- **<missing 4>** should be $ n=4 $ since the factors of $ 3! $ and $ 4! $ are being listed as examples that satisfy the condition.
- **Expression 5** is $ n=4 $.
- So, **<missing 4> = 5**.

- Next:
> *We can also see that $ n=5 $ does not work because the terms **<missing 5>**, and $ 24 $ are consecutive factors of $ 5! $.*

- The terms causing the violation are $ 15 $ and $ 20 $, with $ 24 $.
- **Expression 2** is "15, 20".
- So, **<missing 5> = 2**.

- Then:
> *Also, **<missing 6>** does not work because the terms **<missing 7>**, and $ 9 $ appear consecutively in the factors of **<missing 8>**.*

- **<missing 6>** should be $ n=6 $.
- **Expression 4** is $ n=6 $.
- So, **<missing 6> = 4**.
- **<missing 7>** are the terms $ 6, 8 $.
- **Expression 7** is "6, 8".
- So, **<missing 7> = 7**.
- **<missing 8>** refers to $ 6! $.
- **Expression 20** is $ 6! $.
- So, **<missing 8> = 20**.

2. **Second Paragraph:**

- The solution continues:
> *Note that if we have a prime number **<missing 9>** and an integer **<missing 10>** such that both $ k $ and **<missing 11>** are factors of $ n! $, then the condition cannot be satisfied.*

- **<missing 9>** is " $ p > n $ ".
- **Expression 9** is $ p > n $.
- So, **<missing 9> = 9**.
- **<missing 10>** is " $ k > p $ ".
- **Expression 10** is $ k > p $.
- So, **<missing 10> = 10**.
- **<missing 11>** is " $ k + 1 $ ".
- **Expression 15** is $ k + 1 $.
- So, **<missing 11> = 15**.

3. **Third Paragraph (Odd $ n $):**

- Next, the solution analyzes odd $ n $:
> *If **<missing 12>** is odd, then $ \(2$\left(\frac{n - 1}{2}\right)(n - 1) = n^2 - 2n + 1 \) is a factor of $ n! $. Also, **<missing 13>** is a factor of **<missing 14>**.*

- **<missing 12>** is "$ n \geq 7 $".
- **Expression 3** is $ n \geq 7 $.
- So, **<missing 12> = 3**.
- **<missing 13>** refers to $ (n - 2)(n) = n^2 - 2n $.
- **Expression 12** is $ (n - 2)(n) = n^2 - 2n $.
- So, **<missing 13> = 12**.
- **<missing 14>** is $ n! $.
- **Expression 13** is $ n! $.
- So, **<missing 14> = 13**.

- Then:
> *Since **<missing 15>** for all $ n \geq 7 $, we can use Bertrand's Postulate to show that there is at least one prime number $ p $ such that **<missing 16>**.*

- **<missing 15>** is $ 2n < n^2 - 2n $.
- **Expression 22** is $ 2n < n^2 - 2n $.
- So, **<missing 15> = 22**.
- **<missing 16>** is $ n < p < n^2 - 2n $.
- **Expression 8** is $ n < p < n^2 - 2n $.
- So, **<missing 16> = 8**.

- Finally:
> *Since we have two consecutive factors of $ n! $ and a prime number between the smaller of these factors and $ n $, the condition will not be satisfied for all odd **<missing 17>**.*

- **<missing 17>** is "$ n \geq 7 $".
- **Expression 23** is $ n \geq 7 $.
- So, **<missing 17> = 23**.

4. **Fourth Paragraph (Even $ n $):**

- The solution now considers even $ n $:
> *If **<missing 18>** is even, then **<missing 19>** is a factor of $ n! $. Also, $ (n - 3)(n - 1) = n^2 - 4n + 3 $ is a factor of $ n! $. Since $ 2n < n^2 - 4n + 3 $ for all $ n \geq 8 $, we can use Bertrand's Postulate again to show that there is at least one prime number $ p $ such that $ n < p < n^2 - 4n + 3 $.*

- **<missing 18>** is "$ n \geq 8 $".
- **Expression 18** is $ n \geq 8 $.
- So, **<missing 18> = 18**.
- **<missing 19>** is $ (n - 2)(n) = n^2 - 2n $.
- **Expression 12** is $ (n - 2)(n) = n^2 - 2n $.
- So, **<missing 19> = 12**.

- Then:
> *Since we have two consecutive factors of **<missing 20>** and a prime number between the smaller of these factors and $ n $, the condition will not be satisfied for all even **<missing 21>**.*

- **<missing 20>** is $ n! $.
- **Expression 6** is $ n! $.
- So, **<missing 20> = 6**.
- **<missing 21>** is "$ n \geq 8 $".
- **Expression 21** is $ n \geq 8 $.
- So, **<missing 21> = 21**.

5. **Conclusion:**

- Finally:
> *Therefore, the only numbers that work are **<missing 22>** and **<missing 23>**.*

- **<missing 22>** is $ n=3 $.
- **Expression 11** is $ n=3 $.
- So, **<missing 22> = 11**.
- **<missing 23>** is $ n=4 $.
- **Expression 5** is $ n=4 $.
- So, **<missing 23> = 5**.

**Answer:** 5,2,4,20,1,2,4,7,20,9,10,15,3,12,13,22,8,23,18,12,6,21,11,5

---

**Note:** The numbers correspond to the expression identifiers that fill the missing parts in order. Here's the list of expressions used:

- **5:** $ n = 4 $
- **2:** $ 15, 20 $
- **4:** $ n = 6 $
- **20:** $ 6! $
- **1:** $ n = 5 $
- **7:** $ 6, 8 $
- **9:** $ p > n $
- **10:** $ k > p $
- **15:** $ k + 1 $
- **3:** $ n \geq 7 $
- **12:** $ (n - 2)(n) = n^2 - 2n $
- **13:** $ n! $
- **22:** $ 2n < n^2 - 2n $
- **8:** $ n < p < n^2 - 2n $
- **23:** $ n \geq 7 $
- **18:** $ n \geq 8 $
- **6:** $ n! $
- **21:** $ n \geq 8 $
- **11:** $ n = 3 $

1,14,17,19,2,4,7,20,9,10,15,23,12,6,22,8,3,18,16,13,21,11,5

gpt-4o-2024-08-06

To solve this problem, we need to match the missing parts in the solution with the correct expressions. Let's go through the solution step by step and fill in the missing parts.

1. **First Missing Part:**
- The solution mentions that $n=3$ and $n=4$ work, but $n=5$ does not work. The first missing part is likely referring to $n=5$.
- **Match:** <missing 1> = <expression 1> (n=5)

2. **Second Missing Part:**
- The second missing part is referring to the factorial of the number that does not work, which is $5!$.
- **Match:** <missing 2> = <expression 14> (5!)

3. **Third Missing Part:**
- The third missing part is about $n=6$ not working, as it mentions the terms $6, 8$, and $9$.
- **Match:** <missing 3> = <expression 4> (n=6)

4. **Fourth Missing Part:**
- The fourth missing part is about the numbers that work, which are $n=3$ and $n=4$.
- **Match:** <missing 4> = <expression 5> (n=4)

5. **Fifth Missing Part:**
- The fifth missing part is about the terms $15, 20$, and $24$ for $n=5$.
- **Match:** <missing 5> = <expression 2> (15, 20)

6. **Sixth Missing Part:**
- The sixth missing part is about $n=6$ not working.
- **Match:** <missing 6> = <expression 17> (n=6)

7. **Seventh Missing Part:**
- The seventh missing part is about the terms $6, 8$, and $9$ for $n=6$.
- **Match:** <missing 7> = <expression 7> (6, 8)

8. **Eighth Missing Part:**
- The eighth missing part is about the factorial of $n=6$, which is $6!$.
- **Match:** <missing 8> = <expression 20> (6!)

9. **Ninth Missing Part:**
- The ninth missing part is about a prime number greater than $n$.
- **Match:** <missing 9> = <expression 9> (p>n)

10. **Tenth Missing Part:**
- The tenth missing part is about an integer $k$ such that $k>p$.
- **Match:** <missing 10> = <expression 10> (k>p)

11. **Eleventh Missing Part:**
- The eleventh missing part is about $k+1$ being a factor of $n!$.
- **Match:** <missing 11> = <expression 15> (k+1)

12. **Twelfth Missing Part:**
- The twelfth missing part is about $(n-2)(n)=n^2-2n$ being odd.
- **Match:** <missing 12> = <expression 12> ((n-2)(n)=n^2-2n)

13. **Thirteenth Missing Part:**
- The thirteenth missing part is about $n!$ being a factor.
- **Match:** <missing 13> = <expression 13> (n!)

14. **Fourteenth Missing Part:**
- The fourteenth missing part is about $5!$.
- **Match:** <missing 14> = <expression 14> (5!)

15. **Fifteenth Missing Part:**
- The fifteenth missing part is about $2n<n^2-2n$ for $n\geq7$.
- **Match:** <missing 15> = <expression 22> (2n<n^2-2n)

16. **Sixteenth Missing Part:**
- The sixteenth missing part is about $n<p<n^2-2n$.
- **Match:** <missing 16> = <expression 8> (n<p<n^2-2n)

17. **Seventeenth Missing Part:**
- The seventeenth missing part is about odd $n$.
- **Match:** <missing 17> = <expression 3> (n\geq7)

18. **Eighteenth Missing Part:**
- The eighteenth missing part is about even $n$.
- **Match:** <missing 18> = <expression 18> (n\geq8)

19. **Nineteenth Missing Part:**
- The nineteenth missing part is about $n=4$.
- **Match:** <missing 19> = <expression 19> (n=4)

20. **Twentieth Missing Part:**
- The twentieth missing part is about $6!$.
- **Match:** <missing 20> = <expression 20> (6!)

21. **Twenty-first Missing Part:**
- The twenty-first missing part is about even $n$.
- **Match:** <missing 21> = <expression 21> (n\geq8)

22. **Twenty-second Missing Part:**
- The twenty-second missing part is about the numbers that work, which are $n=3$ and $n=4$.
- **Match:** <missing 22> = <expression 11> (n=3)

23. **Twenty-third Missing Part:**
- The twenty-third missing part is about the numbers that work, which are $n=3$ and $n=4$.
- **Match:** <missing 23> = <expression 5> (n=4)

Now, let's compile the final answer:

Answer: 1, 14, 4, 5, 2, 17, 7, 20, 9, 10, 15, 12, 13, 14, 22, 8, 3, 18, 19, 20, 21, 11, 5

1,14,17,19,2,4,7,20,9,10,15,23,12,6,22,8,3,18,16,13,21,11,5

DeepSeek-Coder-V2-Lite-Instruct

To solve the problem, we need to match the missing tags in the solution with the corresponding formulae. Let's go through the solution step by step and identify the missing parts.

1. **Verification for $ n = 3 $ and $ n = 4 $**:
- The solution mentions that $ n = 3 $ and $ n = 4 $ work.
- This corresponds to the expression identifiers: $ n = 3 $ and $ n = 4 $.

2. **Verification for $ n = 5 $**:
- The solution states that $ n = 5 $ does not work.
- The terms $ 15, 20, $ and $ 24 $ are consecutive factors of $ 5! $.
- This corresponds to the expression identifiers: $ 15, 20 $ and $ 24 $.

3. **Verification for $ n = 6 $**:
- The solution mentions that $ n = 6 $ does not work.
- The terms $ 6, 8, $ and $ 9 $ appear consecutively in the factors of $ 6! $.
- This corresponds to the expression identifiers: $ 6, 8 $ and $ 9 $.

4. **General Case Analysis**:
- The solution discusses the case where $ n $ is odd and even separately.
- For odd $ n $, it mentions that $ (2)(\frac{n-1}{2})(n-1) = n^2 - 2n + 1 $ is a factor of $ n! $.
- This corresponds to the expression identifier: $ (2)(\frac{n-1}{2})(n-1) = n^2 - 2n + 1 $.

5. **Prime Number Analysis**:
- The solution uses Bertrand's Postulate to show that there is at least one prime number $ p $ such that $ n < p < n^2 - 2n $.
- This corresponds to the expression identifier: $ n < p < n^2 - 2n $.

6. **Conclusion**:
- The solution concludes that the only numbers that work are $ n = 2 $ and $ n = 3 $.
- This corresponds to the expression identifiers: $ n = 2 $ and $ n = 3 $.

Based on the above reasoning, the expression identifiers that fill the missing parts are:

- $ n = 3 $ and $ n = 4 $
- $ 15, 20 $ and $ 24 $
- $ n \geq 7 $
- $ n = 6 $
- $ n = 4 $
- $ n! $
- $ 6, 8 $ and $ 9 $
- $ n < p < n^2 - 2n $
- $ p > n $
- $ k > p $
- $ n = 3 $
- $ (n-2)(n) = n^2 - 2n $
- $ n! $
- $ 5! $
- $ k + 1 $
- $ (2)(\frac{n-2}{2})(n-2) = n^2 - 4n + 4 $
- $ n = 6 $
- $ n \geq 8 $
- $ n = 4 $
- $ 6! $
- $ n \geq 8 $
- $ 2n < n^2 - 2n $
- $ n \geq 7 $

Answer: 5, 22, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23

1,14,17,19,2,4,7,20,9,10,15,23,12,6,22,8,3,18,16,13,21,11,5

Meta-Llama-3.1-405B-Instruct-Turbo

To solve this problem, we need to carefully read the solution and match the missing tags with the given formulae.

The first missing tag is <missing 1>, which is described as a value of n that does not work. The corresponding formula is <expression 1>, which is n=5.

The second missing tag is <missing 2>, which is described as the factorial of a number. The corresponding formula is <expression 6>, which is n!.

The third missing tag is <missing 3>, which is described as a value of n that does not work. The corresponding formula is <expression 3>, which is n≥7.

The fourth missing tag is <missing 4>, which is described as a value of n that works. The corresponding formula is <expression 5>, which is n=4.

The fifth missing tag is <missing 5>, which is described as a term in the factors of 5!. The corresponding formula is <expression 2>, which is 15, 20.

The sixth missing tag is <missing 6>, which is described as a value of n that does not work. The corresponding formula is <expression 4>, which is n=6.

The seventh missing tag is <missing 7>, which is described as a term in the factors of 6!. The corresponding formula is <expression 7>, which is 6, 8.

The eighth missing tag is <missing 8>, which is described as the factorial of a number. The corresponding formula is <expression 20>, which is 6!.

The ninth missing tag is <missing 9>, which is described as a prime number. The corresponding formula is <expression 9>, which is p>n.

The tenth missing tag is <missing 10>, which is described as an integer. The corresponding formula is <expression 10>, which is k>p.

The eleventh missing tag is <missing 11>, which is described as a factor of n!. The corresponding formula is <expression 11>, which is n=3.

The twelfth missing tag is <missing 12>, which is described as an odd number. The corresponding formula is <expression 12>, which is (n-2)(n)=n^2-2n.

The thirteenth missing tag is <missing 13>, which is described as a factor of n!. The corresponding formula is <expression 13>, which is n!.

The fourteenth missing tag is <missing 14>, which is described as the factorial of a number. The corresponding formula is <expression 14>, which is 5!.

The fifteenth missing tag is <missing 15>, which is described as an inequality. The corresponding formula is <expression 22>, which is 2n<n^2-2n.

The sixteenth missing tag is <missing 16>, which is described as an inequality. The corresponding formula is <expression 8>, which is n<p<n^2-2n.

The seventeenth missing tag is <missing 17>, which is described as an odd number. The corresponding formula is <expression 17>, which is n=6, but this is incorrect, so we will use <expression 3>, which is n≥7.

The eighteenth missing tag is <missing 18>, which is described as an even number. The corresponding formula is <expression 18>, which is n≥8.

The nineteenth missing tag is <missing 19>, which is described as a factor of n!. The corresponding formula is <expression 19>, which is n=4, but this is incorrect, so we will use <expression 16>, which is (2)(n-2)/2)(n-2)=n^2-4n+4.

The twentieth missing tag is <missing 20>, which is described as the factorial of a number. The corresponding formula is <expression 6>, which is n!.

The twenty-first missing tag is <missing 21>, which is described as an even number. The corresponding formula is <expression 21>, which is n≥8.

The twenty-second missing tag is <missing 22>, which is described as a number that works. The corresponding formula is <expression 11>, which is n=3.

The twenty-third missing tag is <missing 23>, which is described as a number that works. The corresponding formula is <expression 5>, which is n=4.

Answer: 1, 6, 3, 5, 2, 4, 7, 20, 9, 10, 11, 12, 13, 14, 22, 8, 3, 18, 16, 6, 21, 11, 5

1,14,17,19,2,4,7,20,9,10,15,23,12,6,22,8,3,18,16,13,21,11,5

Meta-Llama-3.1-70B-Instruct-Turbo

To solve this problem, we need to carefully match the missing tags in the solution with the given formulae. Let's go through the solution step by step and identify the corresponding formulae.

1. The first missing tag is <missing 1>, which corresponds to the expression "n=5". This is because the solution states that "n=5 does not work". Therefore, <missing 1> = <expression 1>.

2. The next missing tag is <missing 2>, which corresponds to the expression "15, 20". This is because the solution states that "the terms 15, 20, and 24 are consecutive factors of <missing 2>". Therefore, <missing 2> = <expression 2>.

3. The next missing tag is <missing 3>, which corresponds to the expression "n=6". This is because the solution states that "<missing 3> does not work because the terms 6, 8, and 9 appear consecutively in the factors of 6!". Therefore, <missing 3> = <expression 4>.

4. The next missing tag is <missing 4>, which corresponds to the expression "n=4". This is because the solution states that "n=3 and <missing 4> work by listing out the factors of 3! and 4!". Therefore, <missing 4> = <expression 5>.

5. The next missing tag is <missing 5>, which corresponds to the expression "6, 8". This is because the solution states that "n=5 does not work because the terms <missing 5>, and 24 are consecutive factors of 5!". Therefore, <missing 5> = <expression 7>.

6. The next missing tag is <missing 6>, which corresponds to the expression "n=6". This is because the solution states that "<missing 6> does not work because the terms <missing 7>, and 9 appear consecutively in the factors of <missing 8>". Therefore, <missing 6> = <expression 4>.

7. The next missing tag is <missing 7>, which corresponds to the expression "6, 8". This is because the solution states that "<missing 6> does not work because the terms <missing 7>, and 9 appear consecutively in the factors of <missing 8>". Therefore, <missing 7> = <expression 7>.

8. The next missing tag is <missing 8>, which corresponds to the expression "6!". This is because the solution states that "<missing 6> does not work because the terms <missing 7>, and 9 appear consecutively in the factors of <missing 8>". Therefore, <missing 8> = <expression 20>.

9. The next missing tag is <missing 9>, which corresponds to the expression "p>n". This is because the solution states that "if we have a prime number <missing 9> and an integer <missing 10> such that both k and <missing 11> are factors of n!, then the condition cannot be satisfied". Therefore, <missing 9> = <expression 9>.

10. The next missing tag is <missing 10>, which corresponds to the expression "k>p". This is because the solution states that "if we have a prime number <missing 9> and an integer <missing 10> such that both k and <missing 11> are factors of n!, then the condition cannot be satisfied". Therefore, <missing 10> = <expression 10>.

11. The next missing tag is <missing 11>, which corresponds to the expression "n=3". This is because the solution states that "if we have a prime number <missing 9> and an integer <missing 10> such that both k and <missing 11> are factors of n!, then the condition cannot be satisfied". Therefore, <missing 11> = <expression 11>.

12. The next missing tag is <missing 12>, which corresponds to the expression "(n-2)(n)=n^2-2n". This is because the solution states that "if <missing 12> is odd, then (2)(\frac{n-1}{2})(n-1)=n^2-2n+1 is a factor of n!". Therefore, <missing 12> = <expression 12>.

13. The next missing tag is <missing 13>, which corresponds to the expression "n!". This is because the solution states that "<missing 13> is a factor of <missing 14>". Therefore, <missing 13> = <expression 13>.

14. The next missing tag is <missing 14>, which corresponds to the expression "5!". This is because the solution states that "<missing 13> is a factor of <missing 14>". Therefore, <missing 14> = <expression 14>.

15. The next missing tag is <missing 15>, which corresponds to the expression "k+1". This is because the solution states that "since <missing 15> for all n\geq7, we can use Bertrand's Postulate to show that there is at least one prime number p such that <missing 16>". Therefore, <missing 15> = <expression 15>.

16. The next missing tag is <missing 16>, which corresponds to the expression "(2)(\frac{n-2}{2})(n-2)=n^2-4n+4". This is because the solution states that "since <missing 15> for all n\geq7, we can use Bertrand's Postulate to show that there is at least one prime number p such that <missing 16>". Therefore, <missing 16> = <expression 16>.

17. The next missing tag is <missing 17>, which corresponds to the expression "n=6". This is because the solution states that "the condition will not be satisfied for all odd <missing 17>". Therefore, <missing 17> = <expression 17>.

18. The next missing tag is <missing 18>, which corresponds to the expression "n\geq8". This is because the solution states that "if <missing 18> is even, then <missing 19> is a factor of n!". Therefore, <missing 18> = <expression 18>.

19. The next missing tag is <missing 19>, which corresponds to the expression "n=4". This is because the solution states that "if <missing 18> is even, then <missing 19> is a factor of n!". Therefore, <missing 19> = <expression 19>.

20. The next missing tag is <missing 20>, which corresponds to the expression "6!". This is because the solution states that "since 2n<n^2-4n+3 for all n\geq8, we can use Bertrand's Postulate again to show that there is at least one prime number p such that n<p<n^2-4n+3. Since we have two consecutive factors of <missing 20> and a prime number between the smaller of these factors and n, the condition will not be satisfied for all even <missing 21>". Therefore, <missing 20> = <expression 20>.

21. The next missing tag is <missing 21>, which corresponds to the expression "n\geq8". This is because the solution states that "since 2n<n^2-4n+3 for all n\geq8, we can use Bertrand's Postulate again to show that there is at least one prime number p such that n<p<n^2-4n+3. Since we have two consecutive factors of <missing 20> and a prime number between the smaller of these factors and n, the condition will not be satisfied for all even <missing 21>". Therefore, <missing 21> = <expression 21>.

22. The next missing tag is <missing 22>, which corresponds to the expression "2n<n^2-2n". This is because the solution states that "therefore, the only numbers that work are <missing 22> and <missing 23>". Therefore, <missing 22> = <expression 22>.

23. The next missing tag is <missing 23>, which corresponds to the expression "n\geq7". This is because the solution states that "therefore, the only numbers that work are <missing 22> and <missing 23>". Therefore, <missing 23> = <expression 23>.

Answer: 1, 2, 4, 5, 7, 4, 20, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23

1,14,17,19,2,4,7,20,9,10,15,23,12,6,22,8,3,18,16,13,21,11,5

Qwen2-72B-Instruct

The solution mentions that $n=5$ does not work because the terms $15, 20$ , and $24$ are consecutive factors of $5!$. This corresponds to expressions 1, 2, and 14. Then it says that $n=6$ does not work because the terms $6, 8$ , and $9$ appear consecutively in the factors of $6!$. This corresponds to expressions 4, 7, and 20. It then goes on to say that if $n$ is odd and greater than or equal to 7, then the condition cannot be satisfied due to Bertrand's Postulate. This corresponds to expression 3. Similarly, if $n$ is even and greater than or equal to 8, then the condition cannot be satisfied due to Bertrand's Postulate. This corresponds to expression 18. The solution then concludes that the only numbers that work are $n=3$ and $n=4$. This corresponds to expressions 11 and 5.

Answer: 1, 2, 3, 4, 5, 7, 20, 3, 12, 13, 14, 15, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 11, 5

1,14,17,19,2,4,7,20,9,10,15,23,12,6,22,8,3,18,16,13,21,11,5

baichuan4

To solve this problem, we need to match the missing tags in the solution with the provided formulae. Let's go through the solution step by step and identify which formulae fit into the missing tags.

1. The first missing tag is <missing 1>. The context suggests that it should be a value of $n$ that does not work. The only formula that fits this description is <expression 1> which is $n=5$.

2. The second missing tag is <missing 2>. This should be a factorial that has consecutive factors mentioned. The only factorial mentioned in the formulae list is <expression 6> which is $n!$. However, we need to specify which factorial, so we'll keep this in mind and revisit it later.

3. The third missing tag is <missing 3>. This should be a value of $n$ that does not work. The only formula that fits this description and hasn't been used yet is <expression 4> which is $n=6$.

4. The fourth missing tag is <missing 4>. This should be a value of $n$ that works, as it's mentioned along with $n=3$. The only formula that fits this description and hasn't been used yet is <expression 5> which is $n=4$.

5. The fifth missing tag is <missing 5>. This should be a sequence of consecutive factors. The only sequence that fits this description is <expression 2> which is $15, 20$.

6. The sixth missing tag is <missing 6>. This should be a value of $n$ that does not work. The only formula that fits this description and hasn't been used yet is <expression 17> which is $n=6$.

7. The seventh missing tag is <missing 7>. This should be a sequence of consecutive factors. The only sequence that fits this description and hasn't been used yet is <expression 7> which is $6, 8$.

8. The eighth missing tag is <missing 8>. This should be a factorial that has the sequence of factors mentioned in <missing 7>. The only factorial mentioned in the formulae list that hasn't been used yet is <expression 20> which is $6!$.

9. The ninth missing tag is <missing 9>. This should be a prime number condition. The only formula that fits this description is <expression 9> which is $p>n$.

10. The tenth missing tag is <missing 10>. This should be an integer condition related to the prime number mentioned in <missing 9>. The only formula that fits this description is <expression 10> which is $k>p$.

11. The eleventh missing tag is <missing 11>. This should be a value of $n$ that works. The only formula that fits this description and hasn't been used yet is <expression 11> which is $n=3$.

12. The twelfth missing tag is <missing 12>. This should be an expression related to odd values of $n$. The only formula that fits this description is <expression 12> which is $(n-2)(n)=n^2-2n$.

13. The thirteenth missing tag is <missing 13>. This should be a factorial. The only factorial mentioned in the formulae list that hasn't been used yet is <expression 13> which is $n!$.

14. The fourteenth missing tag is <missing 14>. This should be a specific factorial related to the one mentioned in <missing 13>. The only factorial mentioned in the formulae list that hasn't been used yet is <expression 14> which is $5!$.

15. The fifteenth missing tag is <missing 15>. This should be a condition related to consecutive factors. The only formula that fits this description is <expression 15> which is $k+1$.

16. The sixteenth missing tag is <missing 16>. This should be an expression related to even values of $n$. The only formula that fits this description is <expression 16> which is $(2)(\frac{n-2}{2})(n-2)=n^2-4n+4$.

17. The seventeenth missing tag is <missing 17>. This should be a condition related to odd values of $n$. The only formula that fits this description and hasn't been used yet is <expression 23> which is $n\geq7$.

18. The eighteenth missing tag is <missing 18>. This should be a condition related to even values of $n$. The only formula that fits this description is <expression 18> which is $n\geq8$.

19. The nineteenth missing tag is <missing 19>. This should be a specific value of $n$ related to even values. The only formula that fits this description and hasn't been used yet is <expression 19> which is $n=4$.

20. The twentieth missing tag is <missing 20>. This should be a factorial related to even values of $n$. The only factorial mentioned in the formulae list that hasn't been used yet is <expression 20> which is $6!$.

21. The twenty-first missing tag is <missing 21>. This should be a condition related to even values of $n$. The only formula that fits this description and hasn't been used yet is <expression 21> which is $n\geq8$.

22. The twenty-second missing tag is <missing 22>. This should be the final conclusion for values of $n$ that work. The only formula that fits this description and hasn't been used yet is <expression 11> which is $n=3$.

23. The twenty-third missing tag is <missing 23>. This should be the final conclusion for values of $n$ that work. The only formula that fits this description and hasn't been used yet is <expression 5> which is $n=4$.

Therefore, the answer is:

<Detailed reasoning>

Answer: 1, 6, 4, 5, 2, 17, 7, 20, 9, 10, 11, 12, 13, 14, 15, 16, 23, 18, 19, 20, 21, 11, 5

1,14,17,19,2,4,7,20,9,10,15,23,12,6,22,8,3,18,16,13,21,11,5

doubao

<Detailed reasoning>: In the solution, <missing 1> is related to a case that doesn't work, and expression 1 is "n=5" which is the value that doesn't work, so 1 fills <missing 1>. <missing 2> is related to the factors of 5!, and expression 14 is "5!" so 14 fills <missing 2>. <missing 3> is related to a case that doesn't work, and expression 4 is "n=6" which is the value that doesn't work, so 4 fills <missing 3>. <missing 4> is for the other value that works initially, and expression 5 is "n=4" so 5 fills <missing 4>. <missing 5> is related to the consecutive factors of 5!, and expression 2 is "15, 20" so 2 fills <missing 5>. <missing 6> is related to the factors of 6!, and expression 6 is "n!" so 6 fills <missing 6>. <missing 7> is related to the consecutive factors of 6!, and expression 7 is "6, 8" so 7 fills <missing 7>. <missing 8> is related to the factors of a specific value, and expression 20 is "6!" so 20 fills <missing 8>. <missing 9> is related to a prime number condition, and expression 9 is "p>n" so 9 fills <missing 9>. <missing 10> is related to an integer condition with a prime, and expression 10 is "k>p" so 10 fills <missing 10>. <missing 11> is the other value that works initially, and expression 11 is "n=3" so 11 fills <missing 11>. <missing 12> is for an odd case, and expression 12 is "(n-2)(n)=n^2-2n" so 12 fills <missing 12>. <missing 13> is related to a factor, and expression 13 is "n!" so 13 fills <missing 13>. <missing 15> is related to a comparison, and expression 15 is "k+1" so 15 fills <missing 15>. <missing 16> is for a specific calculation, and expression 16 is "(2)(\frac{n-2}{2})(n-2)=n^2-4n+4" so 16 fills <missing 16>. <missing 17

1,14,17,19,2,4,7,20,9,10,15,23,12,6,22,8,3,18,16,13,21,11,5

gpto1

1,14,17,19,2,4,7,20,9,10,15,23,12,6,22,8,3,18,16,13,21,11,5

kimi

<Detailed reasoning>

1. The solution mentions that $n=3$ and $n=4$ work, so <missing 5> should be replaced with <expression 1> n=5.
2. The solution also mentions that $n=5$ does not work because the terms 15, 20 are consecutive factors of $5!$. So <missing 2> should be replaced with <expression 2> 15, 20.
3. The solution states that <missing 3> does not work because the terms 6, 8 appear consecutively in the factors of $6!$. So <missing 3> should be replaced with <expression 4> n=6.
4. The solution mentions that if we have a prime number <missing 9> and an integer <missing 10> such that both $k$ and <missing 11> are factors of $n!$, then the condition cannot be satisfied. Since <missing 9> is a prime number and <missing 10> is an integer, we can assume that <missing 9> is replaced with <expression 9> p>n and <missing 10> is replaced with <expression 10> k>p.
5. The solution mentions that if <missing 12> is odd, then $(2)(\frac{n-1}{2})(n-1)=n^2-2n+1$ is a factor of $n!$. So <missing 12> should be replaced with <expression 12> (n-2)(n)=n^2-2n.
6. The solution mentions that <missing 13> is a factor of <missing 14>. Since <missing 14> is $5!$, we can assume that <missing 13> is replaced with <expression 13> n!.
7. The solution states that since <missing 15> for all $n\geq7$, we can use Bertrand's Postulate to show that there is at least one prime number $p$ such that <missing 16>. Since <missing 15> is $k+1$, we can assume that <missing 15> is replaced with <expression 15> k+1.
8. The solution mentions that if <missing 18> is even, then <missing 19> is a factor of $n!$. Since <missing 18> is even and <missing 19> is a factor of $n!$, we can assume that <missing 18> is replaced with <expression 18> n\geq8 and <missing 19> is replaced with <expression 19> n=4.
9. The solution mentions that since $2n<n^2-4n+3$ for all $n\geq8$, we can use Bertrand's Postulate again to show that there is at least one prime number $p$ such that $n<p<n^2-4n+3$. So <missing 20> should be replaced with <expression 20> 6!.
10. The solution states that the only numbers that work are <missing 22> and <missing 23>. Since <missing 22> is $2n<n^2-2n$ and <missing 23> is $n\geq7$, we can assume that <missing 22> is replaced with <expression 22> 2n<n^2-2n and <missing 23> is replaced with <expression 23> n\geq7.

Answer: 1,2,4,6,9,10,12,13,15,18,19,20,22,23

1,14,17,19,2,4,7,20,9,10,15,23,12,6,22,8,3,18,16,13,21,11,5

minimax

To solve this problem, we need to match the missing parts in the solution with the corresponding expressions provided. Let's go through the solution step by step and match the expressions to the missing tags.

1. The solution starts by verifying that $n=3$ and $n=4$ work. This is straightforward and doesn't require any formulae from the list.

2. The solution then states that $n=5$ does not work because the terms <missing 5> and $24$ are consecutive factors of $5!$. Since $24$ is a factor of $5!$, we need to find a consecutive factor that, when subtracted from $24$, gives a result that would violate the condition. The only consecutive factor of $5!$ that would do this is $20$, because $24 - 20 = 4$ and $20 - 15 = 5$, which does not satisfy the condition $d_2-d_1 \leq d_3-d_2$. Therefore, <missing 5> corresponds to $15$.

3. The solution also mentions that <missing 6> does not work because the terms <missing 7> and $9$ appear

1,14,17,19,2,4,7,20,9,10,15,23,12,6,22,8,3,18,16,13,21,11,5

wenxinyiyan

First, let's analyze the given solution step by step.

1. The solution starts by verifying that $n=3$ and $n=4$ work. This gives us the first hint that <missing 1> could be $n=5$ since it's the next integer to be checked.
2. Next, the solution says "<missing 1> does not work because the terms $15, 20$, and $24$ are consecutive factors of <missing 2>". This confirms that <missing 1> is indeed $n=5$ and <missing 2> is $5!$ because $15, 20, 24$ are consecutive factors of $5!$.
3. The solution then says "<missing 3> does not work because the terms $6, 8$, and $9$ appear consecutively in the factors of $6!$". This tells us that <missing 3> is $n=6$.
4. "<missing 4> work by listing out the factors of $3!$ and $4!$". This means <missing 4> is $n=4$ since it's the only other value besides $n=3$ that was verified to work.
5. "the terms <missing 5>, and $24$ are consecutive factors of $5!$". This tells us that <missing 5> is $15, 20$ since these are the terms mentioned earlier that are consecutive factors of $5!$.
6. "<missing 6> does not work because the terms <missing 7>, and $9$ appear consecutively in the factors of <missing 8>". This means <missing 6> is $n=6$ (again) and <missing 7> is $6, 8$ since these are the terms mentioned. <missing 8> must be $6!$ since it's talking about the factors of $6!$.
7. "If <missing 9> and <missing 10> such that both $k$ and <missing 11> are factors of $n!$". This part is tricky because it's not clear what $k$ is, but since <missing 11> is mentioned in the context of factors, it's likely that <missing 11> is a factor of $n!$ as well. Given the previous information, <missing 11> could be $n=3$ since $3!$ was one of the factorials that worked.
8. "If <missing 12> is odd, then $(2)(\frac{n-1}{2})(n-1)=n^2-2n+1$ is a factor of $n!$". This means <missing 12> is likely talking about $n$ being odd.
9. "<missing 13> is a factor of <missing 14>". Given the context, <missing 13> is likely $n!$ again and <missing 14> is $5!$ since it's talking about factors.
10. "Since <missing 15> for all $n\geq7$". This part is unclear without more context, but it seems to be leading into a discussion about Bertrand's Postulate, so <missing 15> could be related to the conditions needed for the postulate to apply.
11. "we can use Bertrand's Postulate to show that there is at least one prime number $p$ such that <missing 16>". This means <missing 16> is likely a condition related to Bertrand's Postulate, such as $n<p<n^2-2n$.
12. "the condition will not be satisfied for all odd <missing 17>". Given the context, <missing 17> is likely $n$ being odd again.
13. "If <missing 18> is even, then <missing 19> is a factor of $n!$". This means <missing 18> is likely talking about $n$ being even and <missing 19> is a factor of $n!$ when $n$ is even. Given the previous information, <missing 19> could be $(n-2)(n)=n^2-2n$ since it's a factor of $n!$ when $n$ is even.
14. "Also, $(n-3)(n-1)=n^2-4n+3$ is a factor of $n!$". This confirms that <missing 19> is indeed $(n-2)(n)=n^2-2n$ since it's listed as another factor.
15. "Since $2n<n^2-4n+

1,14,17,19,2,4,7,20,9,10,15,23,12,6,22,8,3,18,16,13,21,11,5

zhipu

### Detailed Reasoning

1. **<missing 1>**: The context suggests that we are looking for an integer $ n $ that does not satisfy the given property. The terms $ 15, 20 $, and $ 24 $ are consecutive factors of $ n! $. This implies $ n = 5 $ because these numbers are factors of $ 5! $. Thus, <missing 1> should be $ n = 5 $.
- **Match**: <expression 1>

2. **<missing 2>**: This should be the factorial of the number mentioned in <missing 1>, which is $ 5! $.
- **Match**: <expression 14>

3. **<missing 3>**: We need an integer $ n $ where $ 6, 8 $, and $ 9 $ are consecutive factors of $ n! $. This implies $ n = 6 $ because these numbers are factors of $ 6! $.
- **Match**: <expression 4>

4. **<missing 4>**: This should be the next integer after $ n = 3 $ that satisfies the property, which is $ n = 4 $.
- **Match**: <expression 5>

5. **<missing 5>**: These are the terms that, along with $ 24 $, are consecutive factors of $ 5! $. The terms are $ 15 $ and $ 20 $.
- **Match**: <expression 2>

6. **<missing 6>**: This should be the integer $ n $ where $ 6, 8 $, and $ 9 $ are consecutive factors of $ n! $, which is $ n = 6 $.
- **Match**: <expression 4>

7. **<missing 7>**: These are the terms that, along with $ 9 $, are consecutive factors of $ 6! $. The terms are $ 6 $ and $ 8 $.
- **Match**: <expression 7>

8. **<missing 8>**: This should be the factorial of the number mentioned in <missing 6>, which is $ 6! $.
- **Match**: <expression 20>

9. **<missing 9>**: This is a prime number $ p $.
- **Match**: <expression 9>

10. **<missing 10>**: This is an integer $ k $ such that $ k > p $.
- **Match**: <expression 10>

11. **<missing 11>**: This should be $ k+1 $ since both $ k $ and $ k+1 $ are factors of $ n! $.
- **Match**: <expression 15>

12. **<missing 12>**: This refers to $ n $ being odd.
- **Match**: <expression 11>

13. **<missing 13>**: This should be $ n! $.
- **Match**: <expression 13>

14. **<missing 14>**: This should be $ n! $.
- **Match**: <expression 13>

15. **<missing 15>**: This is a condition for $ n $, which is $ n \geq 7 $.
- **Match**: <expression 3>

16. **<missing 16>**: This is a condition involving a prime number $ p $ such that $ n < p < n^2 - 2n $.
- **Match**: <expression 8>

17. **<missing 17>**: This refers to all odd $ n \geq 7 $.
- **Match**: <expression 23>

18. **<missing 18>**: This refers to $ n $ being even.
- **Match**: <expression 19>

19. **<missing 19>**: This should be $ n! $.
- **Match**: <expression 13>

20. **<missing 20>**: This should be $ n! $.
- **Match**: <expression 13>

21. **<missing 21>**: This refers to all even $ n \geq 8 $.
- **Match**: <expression 21>

22. **<missing 22>**: This is one of the numbers that work, which is $ n = 3 $.
- **Match**: <expression 11>

23. **<missing 23>**: This is the other number that works, which is $ n = 4 $.
- **Match**: <expression 5>

### Answer
1, 14, 4, 5, 2, 4, 7, 20, 9, 10, 15, 11, 13, 13, 3, 8, 23, 19, 13, 13, 21, 11, 5

1,14,17,19,2,4,7,20,9,10,15,23,12,6,22,8,3,18,16,13,21,11,5

deepseek-v3

Error code: 400 - {'error': {'code': 'RequestTimeOut', 'param': None, 'message': 'Request timed out, please try again later.', 'type': 'RequestTimeOut'}, 'id': 'chatcmpl-fc67d279-e3d4-9a34-8f14-31acf5d763f5', 'request_id': 'fc67d279-e3d4-9a34-8f14-31acf5d763f5'}

1,14,17,19,2,4,7,20,9,10,15,23,12,6,22,8,3,18,16,13,21,11,5