Pillai's arithmetical function

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

$P(n)=\sum _{k=1}^n\gcd (k,n)$

or equivalently

$P(n)=\sum _{d\mid n}d\phi (n/d)$

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

it also can be written as

$P(n)=\sum _{d\mid n}d\tau (d)\mu (n/d)$

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.