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A349612 Dirichlet convolution of A342001 [{arithmetic derivative of n}/A003557(n)] with A325126 [Dirichlet inverse of rad(n)].

%I A349612 #22 Jun 05 2022 04:24:51
%S A349612 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,
%T A349612 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,
%U A349612 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
%N A349612 Dirichlet convolution of A342001 [{arithmetic derivative of n}/A003557(n)] with A325126 [Dirichlet inverse of rad(n)].
%H A349612 Antti Karttunen, <a href="/A349612/b349612.txt">Table of n, a(n) for n = 1..20000</a>
%F A349612 a(n) = Sum_{d|n} A342001(d) * A325126(n/d).
%F A349612 If p prime, a(p) = 1. - _Bernard Schott_, Nov 28 2021
%F A349612 Dirichlet g.f.: Sum_{p prime} p^s/((p^s-1)*(p^s+p-1)). - _Sebastian Karlsson_, May 05 2022
%t A349612 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 *)
%o A349612 (PARI)
%o A349612 A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
%o A349612 A003557(n) = (n/factorback(factorint(n)[, 1]));
%o A349612 A342001(n) = (A003415(n) / A003557(n));
%o A349612 A007947(n) = factorback(factorint(n)[, 1]); \\ From A007947
%o A349612 memoA325126 = Map();
%o A349612 A325126(n) = if(1==n,1,my(v); if(mapisdefined(memoA325126,n,&v), v, v = -sumdiv(n,d,if(d<n,A007947(n/d)*A325126(d),0)); mapput(memoA325126,n,v); (v)));
%o A349612 A349612(n) = sumdiv(n,d,A342001(d)*A325126(n/d));
%Y A349612 Cf. A003415, A003557, A007947, A342001, A325126.
%Y A349612 Cf. also A349394, A349396, A349618.
%K A349612 sign
%O A349612 1,25
%A A349612 _Antti Karttunen_, Nov 23 2021