Pillai's arithmetical function

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

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

or equivalently^[1]

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

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

it also can be written as^[2]

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

where, $\sigma$ 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]

References

1. Lászlo Tóth (2010). "A survey of gcd-sum functions". J. Integer Sequences. 13.
2. http://math.stackexchange.com/questions/135351/sum-of-gcdk-n
3. S. S. Pillai (1933). "On an arithmetic function". Annamalai University Journal. II: 242–248.

 A018804