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A347135 a(n) = Sum_{d|n} A001615(n/d) * A069359(d).

%I A347135 #14 Oct 28 2021 16:41:07
%S A347135 0,1,1,5,1,12,1,16,7,16,1,51,1,20,18,44,1,68,1,71,22,28,1,156,11,32,
%T A347135 33,91,1,167,1,112,30,40,26,277,1,44,34,220,1,215,1,131,110,52,1,420,
%U A347135 15,140,42,151,1,300,34,284,46,64,1,673,1,68,138,272,38,311,1,191,54,295,1,836,1,80,162,211,38,359,1,596
%N A347135 a(n) = Sum_{d|n} A001615(n/d) * A069359(d).
%C A347135 Dirichlet convolution of A001615 (Dedekind psi function) with A069359.
%C A347135 Dirichlet convolution of A001221 (omega, number of distinct prime factors of n) with A322577.
%H A347135 Antti Karttunen, <a href="/A347135/b347135.txt">Table of n, a(n) for n = 1..10000</a>
%H A347135 Antti Karttunen, <a href="/A347135/a347135.txt">Data supplement: n, a(n) computed for n = 1..65537</a>
%H A347135 Wikipedia, <a href="https://en.wikipedia.org/wiki/Dirichlet_convolution">Dirichlet convolution</a>
%F A347135 a(n) = Sum_{d|n} A001615(n/d) * A069359(d).
%F A347135 a(n) = Sum_{d|n} A001221(n/d) * A322577(d).
%t A347135 Table[DivisorSum[n,PrimeNu[n/#]*Sum[DirichletConvolve[j,MoebiusMu[j]^2,j,#/d] EulerPhi[d],{d,Divisors[#]}]&],{n,80}] (* _Giorgos Kalogeropoulos_, Oct 28 2021 *)
%o A347135 (PARI)
%o A347135 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 A347135 A069359(n) = (n*sumdiv(n, d, isprime(d)/d)); \\ From A069359
%o A347135 A347135(n) = sumdiv(n,d,A001615(n/d)*A069359(d));
%Y A347135 Cf. A001221, A001615, A069359, A322577.
%Y A347135 Cf. also A347132, A347133, A347134.
%K A347135 nonn
%O A347135 1,4
%A A347135 _Antti Karttunen_, Aug 23 2021