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A349380 Dirichlet convolution of A003415 (arithmetic derivative of n) with A349134 (Dirichlet inverse of Kimberling's paraphrases).

%I A349380 #16 Mar 01 2022 09:33:35
%S A349380 0,1,1,3,1,2,1,8,4,3,1,5,1,4,3,20,1,6,1,8,4,6,1,12,7,7,14,11,1,3,1,48,
%T A349380 6,9,5,14,1,10,7,20,1,4,1,17,8,12,1,28,10,13,9,20,1,18,7,28,10,15,1,6,
%U A349380 1,16,11,112,8,6,1,26,12,5,1,32,1,19,11,29,8,7,1,48,46,21,1,8,10,22,15,44,1,6,9,35,16
%N A349380 Dirichlet convolution of A003415 (arithmetic derivative of n) with A349134 (Dirichlet inverse of Kimberling's paraphrases).
%C A349380 Dirichlet convolution of A349394 with A349432.
%C A349380 Dirichlet convolution with A349136 gives A300251.
%H A349380 Antti Karttunen, <a href="/A349380/b349380.txt">Table of n, a(n) for n = 1..20000</a>
%F A349380 a(n) = Sum_{d|n} A003415(n/d) * A349134(d).
%F A349380 a(n) = Sum_{d|n} A349394(n/d) * A349432(d).
%o A349380 (PARI)
%o A349380 A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
%o A349380 A003602(n) = (1+(n>>valuation(n,2)))/2;
%o A349380 memoA349134 = Map();
%o A349380 A349134(n) = if(1==n,1,my(v); if(mapisdefined(memoA349134,n,&v), v, v = -sumdiv(n,d,if(d<n,A003602(n/d)*A349134(d),0)); mapput(memoA349134,n,v); (v)));
%o A349380 A349380(n) = sumdiv(n,d,A003415(d)*A349134(n/d));
%Y A349380 Cf. A003415, A003602, A349134, A349394, A349432.
%Y A349380 Cf. also A300251, A347955, A349136, A349431.
%K A349380 sign,look
%O A349380 1,4
%A A349380 _Antti Karttunen_, Nov 21 2021