Pillai's Arithmetical Function

In number theory, the gcd-sum function,
also called Pillai's arithmetical function, is defined for every $n$ by

:$P(n)=\sum_^n\gcd(k,n)$

or equivalently

:$P(n) = \sum_ d \varphi(n/d)$

where $d$ is a divisor of $n$ and $\varphi$ is Euler's totient function.

it also can be written as

:$P(n) = \sum_ d \tau(d) \mu(n/d)$

where, $\tau$ is the divisor function, and $\mu$ is the Möbius function.

This multiplicative arithmetical function was introduced by the Indian mathematician Subbayya Sivasankaranarayana Pillai in 1933.

References

{{oeis, A018804

Arithmetic functions