cp's OEIS Frontend

A347132 a(n) = Sum_{d|n} A001615(n/d) * A003415(d), where A003415 is the arithmetic derivative and A001615 is Dedekind psi function.

%I A347132 #16 Oct 19 2021 14:21:27
%S A347132 0,1,1,7,1,12,1,30,10,16,1,65,1,20,18,104,1,83,1,93,22,28,1,254,16,32,
%T A347132 63,121,1,167,1,320,30,40,26,391,1,44,34,374,1,215,1,177,143,52,1,840,
%U A347132 22,165,42,205,1,450,34,494,46,64,1,827,1,68,183,912,38,311,1,261,54,295,1,1430,1,80,197,289,38,359
%N A347132 a(n) = Sum_{d|n} A001615(n/d) * A003415(d), where A003415 is the arithmetic derivative and A001615 is Dedekind psi function.
%C A347132 Dirichlet convolution of Dedekind psi function (A001615) with the arithmetic derivative (A003415).
%H A347132 Antti Karttunen, <a href="/A347132/b347132.txt">Table of n, a(n) for n = 1..10000</a>
%H A347132 Antti Karttunen, <a href="/A347132/a347132.txt">Data supplement: n, a(n) computed for n = 1..65537</a>
%F A347132 a(n) = Sum_{d|n} A001615(n/d) * A003415(d).
%t A347132 Table[DivisorSum[n, DirichletConvolve[j, MoebiusMu[j]^2, j, n/#]*If[# < 2, 0, # Total[#2/#1 & @@@ FactorInteger[#]]] &], {n, 78}] (* _Michael De Vlieger_, Oct 19 2021, after _Jan Mangaldan_ at A001615 *)
%o A347132 (PARI)
%o A347132 A001615(n) = if(1==n,n, my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1))); \\ After code in A001615
%o A347132 A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
%o A347132 A347132(n) = sumdiv(n,d,A001615(n/d)*A003415(d));
%Y A347132 Cf. A001615, A003415.
%Y A347132 Cf. also A305809, A322577, A347131, A347133, A347135.
%K A347132 nonn
%O A347132 1,4
%A A347132 _Antti Karttunen_, Aug 23 2021