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%I #23 May 28 2017 09:16:27
%S 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,
%T 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,
%U 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
%N Reciprocals are the complement to logarithm of Riemann zeta. a(1)=0, for n>1: a(n) = A008683(n) + A100995(n).
%C 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.
%H Antti Karttunen, <a href="/A193056/b193056.txt">Table of n, a(n) for n = 1..10000</a>
%F a(1)=0, for n > 1: a(n) = A008683(n) + A100995(n).
%F 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 - ...
%F 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...
%e 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...
%t 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 *)
%o (PARI) A193056(n) = if(1==n,0,moebius(n)+isprimepower(n)); \\ _Antti Karttunen_, May 27 2017
%Y Cf. A008683, A010051, A100995, A175992.
%K sign
%O 1,4
%A _Mats Granvik_, Jul 15 2011
%E Data section extended to 120 terms by _Antti Karttunen_, May 27 2017
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