A304820 -id:A304820 - cp's OEIS frontend

A304817 Number of divisors of n that are either 1 or not a perfect power.

1, 2, 2, 2, 2, 4, 2, 2, 2, 4, 2, 5, 2, 4, 4, 2, 2, 5, 2, 5, 4, 4, 2, 6, 2, 4, 2, 5, 2, 8, 2, 2, 4, 4, 4, 6, 2, 4, 4, 6, 2, 8, 2, 5, 5, 4, 2, 7, 2, 5, 4, 5, 2, 6, 4, 6, 4, 4, 2, 11, 2, 4, 5, 2, 4, 8, 2, 5, 4, 8, 2, 8, 2, 4, 5, 5, 4, 8, 2, 7, 2, 4, 2, 11, 4, 4

Offset: 1

Gus Wiseman, May 18 2018

nonn

First differs from A183095 at a(80) = 8, A183095(80) = 7.

			The a(72) = 8 divisors of 72 that are either 1 or not a perfect power are {1, 2, 3, 6, 12, 18, 24, 72}. Missing are {4, 8, 9, 36}.

Cf. A000005, A001597, A007916, A091050, A183096, A304326, A304362, A304653, A304779, A304819, A304820.

Table[DivisorSum[n,Boole[GCD@@FactorInteger[#][[All,2]]==1]&],{n,100}]

a(n) = sumdiv(n, d, !ispower(d)); \\ Michel Marcus, May 19 2018

a(n) = A183096(n) + 1.

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.

Original entry on oeis.org

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

Gus Wiseman, May 19 2018

sign

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

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

Positions of nonzero entries appear to be A126706.

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

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

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

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

cp's OEIS Frontend

A304817 Number of divisors of n that are either 1 or not a perfect power.

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