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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]

\( P(n) = \sum_{d \mid n} d \sigma(d) \mu(n/d) \)

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

Lászlo Tóth (2010). "A survey of gcd-sum functions". J. Integer Sequences 13.
http://math.stackexchange.com/questions/135351/sum-of-gcdk-n

S. S. Pillai (1933). "On an arithmetic function". Annamalai University Journal II: 242–248.

OEIS A018804

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