text: ### ljsh_king's blog
  
  By ljsh_king, history, 6 weeks ago,
  
  I was reading this article, and saw example 1 (number of coprime pairs in range [1, n]). I tried to solve this problem with similar formula. Here's what I wrote:
  
  $\sum_{i=1}^n \sum_{j=1}^n [gcd(a_i, a_j) = 1]$
  
  $= \sum_{i=1}^n \sum_{j=1}^n \sum_{d|gcd(a_i, a_j)} \mu{(d)}$
  
  $= \sum_{i=1}^n \sum_{j=1}^n \sum_{d=1}^{\infty} [d|a_i][d|a_j]\mu{(d)}$
  
  $= \sum_{d=1}^{\infty} \mu{(d)} (\sum_{i=1}^n [d|a_i]) (\sum_{j=1}^n [d|a_j])$
  
  $= \sum_{d=1}^{\infty} \mu{(d)} (\sum_{i=1}^n [d|a_i])^2$
  
  We can calculate $\mu{(d)}$ using linear sieve in $O(max(a_i))$
  
  and $\sum_{i=1}^n [d|a_i]$ for each $d$ in $O(n * \sqrt{max(a_i)})$
  
  But it turns out that when the input is
  
  2
  1 2
  
  
  mu = {1, -1}
  
  cnt = {2, 1}
  
  Gives output of $1 * 2^2 + -1 * 1^2 = 3$ which is absolutely wrong.
  
  Can anyone point out my mistake? Thanks.
  
  • +8
  
   » 6 weeks ago, # |   +7 The $3$ pairs are $(1, 1), (1, 2), (2, 1)$. It's relatively easy to fix the code in such a way that the order of $i, j$ doesn't matter.
  •  » » 6 weeks ago, # ^ |   +3 That make sense. Thanks for pointing that out!
explanation: The text extract is a blog post discussing a mathematical problem related to counting coprime pairs in a given range. The author presents a formula and an example input, and asks for help in identifying a mistake in their approach. The post includes a question and a response from another user pointing out the issue.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: ### ljsh_king's blog
  
  By ljsh_king, history, 6 weeks ago,
  
  I was reading this article, and saw example 1 (number of coprime pairs in range [1, n]). I tried to solve this problem with similar formula. Here's what I wrote:
  
  $\sum_{i=1}^n \sum_{j=1}^n [gcd(a_i, a_j) = 1]$
  
  $= \sum_{i=1}^n \sum_{j=1}^n \sum_{d|gcd(a_i, a_j)} \mu{(d)}$
  
  $= \sum_{i=1}^n \sum_{j=1}^n \sum_{d=1}^{\infty} [d|a_i][d|a_j]\mu{(d)}$
  
  $= \sum_{d=1}^{\infty} \mu{(d)} (\sum_{i=1}^n [d|a_i]) (\sum_{j=1}^n [d|a_j])$
  
  $= \sum_{d=1}^{\infty} \mu{(d)} (\sum_{i=1}^n [d|a_i])^2$
  
  We can calculate $\mu{(d)}$ using linear sieve in $O(max(a_i))$
  
  and $\sum_{i=1}^n [d|a_i]$ for each $d$ in $O(n * \sqrt{max(a_i)})$
  
  But it turns out that when the input is
  
  2
  1 2
  
  
  mu = {1, -1}
  
  cnt = {2, 1}
  
  Gives output of $1 * 2^2 + -1 * 1^2 = 3$ which is absolutely wrong.
  
  Can anyone point out my mistake? Thanks.
  
  • +8
  
   » 6 weeks ago, # |   +7 The $3$ pairs are $(1, 1), (1, 2), (2, 1)$. It's relatively easy to fix the code in such a way that the order of $i, j$ doesn't matter.
  •  » » 6 weeks ago, # ^ |   +3 That make sense. Thanks for pointing that out!
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The text extract is a blog post discussing a mathematical problem related to counting coprime pairs in a given range. The author presents a formula and an example input, and asks for help in identifying a mistake in their approach. The post includes a question and a response from another user pointing out the issue.
  
  Label: question-and-answer