A304819 Dirichlet convolution of r with zeta, where r(n) = (-1)^Omega(n) if n is 1 or not a perfect power and r(n) = 0 otherwise.

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, -1, 0, -1, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, -1, -1, 0, 0, -1, 0, -1, 0, -1, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, -1, 0, 0, 0, 0, -1, 0, 0, 0, -2, 0, 0, -1, -1, 0, 0, 0, -1, 0

Offset: 1

Author

Gus Wiseman, May 19 2018

Keywords

• sign

Comments

Omega(n) = A001222(n) is the number of prime factors of n counted with multiplicity.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537

Crossrefs

Positions of nonzero entries appear to be A126706.

Cf. A000005, A000007, A001222, A001597, A005117, A007916, A008683, A008836, A304362, A304653, A304779, A304817, A304820.

Programs

Mathematica

Table[Sum[(-1)^PrimeOmega[d],{d,Select[Divisors[n],GCD@@FactorInteger[#][[All,2]]==1&]}],{n,100}]

PARI

A304819(n) = sumdiv(n,d,if(!ispower(d),(-1)^bigomega(d),0)); \\ Antti Karttunen, Jul 29 2018

Formula

a(n) = Sum_{d|n, d = 1 or not a perfect power} (-1)^Omega(d).