Pillai's arithmetical function

In number theory, the gcd-sum function,{{cite journal |author=Lászlo Tóth |title=A survey of gcd-sum functions |journal=J. Integer Sequences |volume=13 |year=2010}}

also called Pillai's arithmetical function, is defined for every $n$ by

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

or equivalently

:$P(n)\; =\; \backslash sum\_\{d\backslash mid\; n\}\; d\; \backslash varphi(n/d)$

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

it also can be written as[https://math.stackexchange.com/q/135351 Sum of GCD(k,n)]

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

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

This multiplicative arithmetical function was introduced by the Indian mathematician Subbayya Sivasankaranarayana Pillai in 1933.{{cite journal |author=S. S. Pillai |title=On an arithmetic function |journal=Annamalai University Journal |volume=II |year=1933 |pages=242–248}}

{{cite journal |last1=Broughan |first1=Kevin |title=The gcd-sum function |journal=Journal of Integer Sequences |date=2002 |volume=4 |issue=Article 01.2.2 |pages=1–19 }}