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 A193056 Reciprocals are the complement to logarithm of Riemann zeta. a(1)=0, for n>1: a(n) = A008683(n) + A100995(n). 2
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 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: A082513 A187495 A187496 * A244417 A086780 A158612 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

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