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A347961 Dirichlet convolution of A342001 with itself.

%I A347961 #15 Dec 24 2024 07:26:20
%S A347961 0,0,0,1,0,2,0,4,1,2,0,14,0,2,2,10,0,14,0,18,2,2,0,42,1,2,4,22,0,40,0,
%T A347961 20,2,2,2,63,0,2,2,58,0,48,0,30,20,2,0,92,1,18,2,34,0,40,2,74,2,2,0,
%U A347961 204,0,2,24,35,2,64,0,42,2,56,0,162,0,2,20,46,2,72,0,132,10,2,0,260,2,2,2,106,0,210,2,54,2,2,2
%N A347961 Dirichlet convolution of A342001 with itself.
%H A347961 Antti Karttunen, <a href="/A347961/b347961.txt">Table of n, a(n) for n = 1..10000</a>
%H A347961 Vaclav Kotesovec, <a href="/A347961/a347961.jpg">Graph - the asymptotic ratio (100000 terms)</a>
%F A347961 a(n) = Sum_{d|n} A342001(d) * A342001(n/d).
%F A347961 From _Vaclav Kotesovec_, Mar 04 2023: (Start)
%F A347961 Let pr(s) = Product_{primes p} (1 + p^(1-2*s) - p^(2-2*s) - p^(-s))
%F A347961 and su(s) = Sum_{primes p} p^s/((p^s - 1)*(p^s + p - 1)).
%F A347961 Sum_{k=1..n} a(k) ~ pr(2)^2 * su(2)^2 * Pi^4 * n^2 * log(n) / 72 *
%F A347961 (1 + (2*gamma - 1/2 + 2*pr'(2)/pr(2) + 2*su'(2)/su(2) + 12*zeta'(2)/Pi^2) / log(n)), where
%F A347961 pr(2) = A065464 = 0.428249505677094440218765707581823546121298513355936...
%F A347961 pr'(2) = pr(2) * Sum_{primes p} (3*p - 2) * log(p) / (p^3 - 2*p + 1) = 0.6293283828324697510445630056425352981207558777167836747744750359407...
%F A347961 su(2) = Sum_{j>=2} (1/2 + (-1)^j * (Fibonacci(j) - 1/2)) * PrimeZetaP(j) = 0.4526952873143153104685540856936425315834753528741817723313791528384...
%F A347961 su'(2) = Sum_{primes p} p^2 * (1-p-p^4) * log(p) / ((p^2-1)^2 * (p^2+p-1)^2) = -0.486606220169261905698805096547122238460686354267440350206456696497...
%F A347961 and gamma is the Euler-Mascheroni constant A001620. (End)
%o A347961 (PARI)
%o A347961 A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
%o A347961 A003557(n) = (n/factorback(factorint(n)[, 1]));
%o A347961 A342001(n) = (A003415(n) / A003557(n));
%o A347961 A347961(n) = sumdiv(n,d,A342001(n/d)*A342001(d));
%Y A347961 Cf. A003415, A003557, A342001, A347962.
%Y A347961 Cf. also A305809, A347963.
%K A347961 nonn
%O A347961 1,6
%A A347961 _Antti Karttunen_, Sep 24 2021