Pillai's arithmetical function explained

In number theory, the gcd-sum function,^[1] also called Pillai's arithmetical function,^[1] is defined for every

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

or equivalently^[1]

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

where

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

it also can be written as^[2]

P(n) = \sum _{d} 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.^[3]

^[4]

Notes and References

1. Lászlo Tóth . A survey of gcd-sum functions . J. Integer Sequences . 13 . 2010.
2. https://math.stackexchange.com/q/135351 Sum of GCD(k,n)
3. S. S. Pillai . On an arithmetic function . Annamalai University Journal . II . 1933 . 242–248.
4. Broughan . Kevin . The gcd-sum function . Journal of Integer Sequences . 2002 . 4 . Article 01.2.2 . 1–19 .