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 A349612 Dirichlet convolution of A342001 [{arithmetic derivative of n}/A003557(n)] with A325126 [Dirichlet inverse of rad(n)]. 4
 0, 1, 1, 0, 1, 0, 1, 1, -1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, -3, 0, 3, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, -5, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, -5, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,25 LINKS Antti Karttunen, Table of n, a(n) for n = 1..20000 FORMULA a(n) = Sum_{d|n} A342001(d) * A325126(n/d). If p prime, a(p) = 1. - Bernard Schott, Nov 28 2021 Dirichlet g.f.: Sum_{p prime} p^s/((p^s-1)*(p^s+p-1)). - Sebastian Karlsson, May 05 2022 MATHEMATICA f[p_, e_] := e/p; d[1] = 0; d[n_] := n * Plus @@ f @@@ FactorInteger[n]; f1[p_, e_] := p^(e-1); s1[1] = 1; s1[n_] := Times @@ f1 @@@ FactorInteger[n]; f2[p_, e_] := -p*(1 - p)^(e - 1); s2[1] = 1; s2[n_] := Times @@ f2 @@@ FactorInteger[n]; a[n_] := DivisorSum[n, d[#]*s2[n/#]/s1[#] &]; Array[a, 100] (* Amiram Eldar, Nov 23 2021 *) PROG (PARI) A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1])); A003557(n) = (n/factorback(factorint(n)[, 1])); A342001(n) = (A003415(n) / A003557(n)); A007947(n) = factorback(factorint(n)[, 1]); \\ From A007947 memoA325126 = Map(); A325126(n) = if(1==n, 1, my(v); if(mapisdefined(memoA325126, n, &v), v, v = -sumdiv(n, d, if(d

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Last modified February 3 14:36 EST 2023. Contains 360035 sequences. (Running on oeis4.)