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

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

or equivalently^[1]

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]

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.^[3]

References

^ ^a ^b ^c Lászlo Tóth (2010). "A survey of gcd-sum functions". J. Integer Sequences. 13.
^ Sum of GCD(k,n)
^ S. S. Pillai (1933). "On an arithmetic function". Annamalai University Journal. II: 242–248.
^ Broughan, Kevin (2002). "The gcd-sum function". Journal of Integer Sequences. 4 (Article 01.2.2): 1–19.