Pillai's arithmetical function - Alchetron, the free social encyclopedia

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

P ( n ) = ∑ k = 1 n gcd ( k , n )

or equivalently

P ( n ) = ∑ d ∣ n d φ ( n / d )

where d is a divisor of n and φ is Euler's totient function.

it also can be written as

P ( n ) = ∑ d ∣ n d σ ( d ) μ ( n / d )

where, σ is the Divisor function, and μ is the Möbius function.

This multiplicative arithmetical function was introduced by the Indian mathematician Subbayya Sivasankaranarayana Pillai in 1933.

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