Pillai's arithmetical function
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In number theory, the gcd-sum function,[1] also called Pillai's arithmetical function,[1] is defined for every [math]\displaystyle{ n }[/math] by
- [math]\displaystyle{ P(n)=\sum_{k=1}^n\gcd(k,n) }[/math]
or equivalently[1]
- [math]\displaystyle{ P(n) = \sum_{d\mid n} d \varphi(n/d) }[/math]
where [math]\displaystyle{ d }[/math] is a divisor of [math]\displaystyle{ n }[/math] and [math]\displaystyle{ \varphi }[/math] is Euler's totient function.
it also can be written as[2]
- [math]\displaystyle{ P(n) = \sum_{d \mid n} d \tau(d) \mu(n/d) }[/math]
where, [math]\displaystyle{ \tau }[/math] is the divisor function, and [math]\displaystyle{ \mu }[/math] is the Möbius function.
This multiplicative arithmetical function was introduced by the Indian mathematician Subbayya Sivasankaranarayana Pillai in 1933.[3]
References
- ↑ 1.0 1.1 1.2 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.
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