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
    $\begingroup$ 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.) $\endgroup$ – Dilip Sarwate Apr 22 '12 at 16:03
  • $\begingroup$ n $\leq$ 200000. Ok I will read Euler's totient function. $\endgroup$ – Elmi Ahmadov Apr 22 '12 at 16:32

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.


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