A193056 Reciprocals are the complement to logarithm of Riemann zeta. a(1)=0, for n>1: a(n) = A008683(n) + A100995(n).

0, 0, 0, 2, 0, 1, 0, 3, 2, 1, 0, 0, 0, 1, 1, 4, 0, 0, 0, 0, 1, 1, 0, 0, 2, 1, 3, 0, 0, -1, 0, 5, 1, 1, 1, 0, 0, 1, 1, 0, 0, -1, 0, 0, 0, 1, 0, 0, 2, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 6, 1, -1, 0, 0, 1, -1, 0, 0, 0, 1, 0, 0, 1, -1, 0, 0, 4, 1, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, -1, 0, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, -1, 1, 0, 0, 1, 1, 0

Offset: 1

Author

Mats Granvik, Jul 15 2011

Keywords

• sign

Comments

The characteristic function of primes can be computed as: A010051(n) = A100995(n) - sqrt(A100995(n)*a(n)). But the element-wise multiplication of the sequences inside the sqrt, has no known operation or definition in terms of Dirichlet generating functions.

Examples

			The reciprocals of this sequence, defined by the Dirichlet series generating function are: 0/1,0/1,0/1,1/2,0/1,1/1,0/1,1/3,1/2,1/1, 0/1,0/1...
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A008683, A010051, A100995, A175992.

Programs

Mathematica

a100995[n_]:=If[PrimePowerQ[n], FactorInteger[n][[1, 2]], 0] (* From Harvey P. Dale *); Table[If[n==1, 0, MoebiusMu[n] + a100995[n]], {n, 100}] (* Indranil Ghosh, May 27 2017 *)

PARI

A193056(n) = if(1==n,0,moebius(n)+isprimepower(n)); \\ Antti Karttunen, May 27 2017

Formula

a(1)=0, for n > 1: a(n) = A008683(n) + A100995(n).

Dirichlet series generating function of reciprocals: -0/1*(Zeta(s)-1)^1 + 1/2*(Zeta(s)-1)^2 - 2/3*(Zeta(s)-1)^3 + 3/4*(Zeta(s)-1)^4 - ...

Reciprocals of a(n) = first column in the sum of matrix powers: -0/1*A175992^1 + 1/2*A175992^2 - 2/3*A175992^3 + 3/4*A175992^4...

Extensions

Data section extended to 120 terms by Antti Karttunen, May 27 2017