I want to find this

$$ \sum_{k=1}^n \gcd(k,n)$$

but I don't know how to solve. Does anybody can help me to finding this problem.


  • 1
    Do you know how many integers $k$ in the range $1 \leq k \leq n$ are relatively prime to $n$ so that $\gcd(k,n) = 1$? (Hint: Read about Euler's totient function.) – Dilip Sarwate Apr 22 '12 at 16:03
  • n $\leq$ 200000. Ok I will read Euler's totient function. – Elmi Ahmadov Apr 22 '12 at 16:32
up vote 16 down vote accepted

This is Pillai's arithmetical function as in OEIS A018804

Formulae given there include $$\sum_{d|n} d \,\phi(n/d)$$ and $$\sum_{d|n} d \, \tau(d) \, \mu(n/d)$$ where $\phi(n)$ is Euler's totient function, $\tau(n)$ is the number of divisors and $\mu(n)$ is the Möbius function.

Your Answer


By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Not the answer you're looking for? Browse other questions tagged or ask your own question.