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

1,4

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.

Links

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

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...

Example

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...

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 *)

Prog

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

Crossrefs

Cf. A008683, A010051, A100995, A175992.

Sequence in context: A187495 A187496 A352552 * A244417 A324811 A086780

Adjacent sequences: A193053 A193054 A193055 * A193057 A193058 A193059

Keyword

sign

Author

Mats Granvik, Jul 15 2011

Extensions

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

Status

approved