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A349619 Dirichlet convolution of A003415 with the Dirichlet inverse of A003557.

%I A349619 #12 Nov 26 2021 10:48:59
%S A349619 0,1,1,3,1,3,1,7,5,5,1,7,1,7,6,15,1,9,1,13,8,11,1,15,9,13,19,19,1,14,
%T A349619 1,31,12,17,10,17,1,19,14,29,1,20,1,31,24,23,1,31,13,25,18,37,1,27,14,
%U A349619 43,20,29,1,30,1,31,34,63,16,32,1,49,24,34,1,33,1,37,34,55,16,38,1,61,65,41,1,44,20,43,30,71
%N A349619 Dirichlet convolution of A003415 with the Dirichlet inverse of A003557.
%H A349619 Antti Karttunen, <a href="/A349619/b349619.txt">Table of n, a(n) for n = 1..20000</a>
%F A349619 a(n) = Sum_{d|n} A003415(n/d) * A349340(d).
%t A349619 f1[p_, e_] := e/p; d[1] = 0; d[n_] := n*Plus @@ f1 @@@ FactorInteger[n]; f[p_, e_] := -(p - 1)^(e - 1); s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; a[n_] := DivisorSum[n, s[#]*d[n/#] &]; Array[a, 100] (* _Amiram Eldar_, Nov 25 2021 *)
%o A349619 (PARI)
%o A349619 A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
%o A349619 A349340(n) = { my(f=factor(n)); prod(i=1, #f~, -((f[i,1]-1)^(f[i,2]-1))); };
%o A349619 A349619(n) = sumdiv(n,d,A003415(n/d)*A349340(d));
%Y A349619 Cf. A003415, A003557, A349340.
%Y A349619 Cf. also A349394, A349618, A349620, A349621.
%K A349619 nonn
%O A349619 1,4
%A A349619 _Antti Karttunen_, Nov 25 2021