A347961 Dirichlet convolution of A342001 with itself.

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, 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, 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

Offset: 1

Author

Antti Karttunen, Sep 24 2021

Keywords

• nonn

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000
• Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)

Crossrefs

Cf. A003415, A003557, A342001, A347962.

Cf. also A305809, A347963.

Programs

PARI

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A003557(n) = (n/factorback(factorint(n)[, 1]));
A342001(n) = (A003415(n) / A003557(n));
A347961(n) = sumdiv(n,d,A342001(n/d)*A342001(d));

Formula

a(n) = Sum_{d|n} A342001(d) * A342001(n/d).

From Vaclav Kotesovec, Mar 04 2023: (Start)

Let pr(s) = Product_{primes p} (1 + p^(1-2*s) - p^(2-2*s) - p^(-s))

and su(s) = Sum_{primes p} p^s/((p^s - 1)*(p^s + p - 1)).

Sum_{k=1..n} a(k) ~ pr(2)^2 * su(2)^2 * Pi^4 * n^2 * log(n) / 72 *

(1 + (2*gamma - 1/2 + 2*pr'(2)/pr(2) + 2*su'(2)/su(2) + 12*zeta'(2)/Pi^2) / log(n)), where

pr(2) = A065464 = 0.428249505677094440218765707581823546121298513355936...

pr'(2) = pr(2) * Sum_{primes p} (3*p - 2) * log(p) / (p^3 - 2*p + 1) = 0.6293283828324697510445630056425352981207558777167836747744750359407...

su(2) = Sum_{j>=2} (1/2 + (-1)^j * (Fibonacci(j) - 1/2)) * PrimeZetaP(j) = 0.4526952873143153104685540856936425315834753528741817723313791528384...

su'(2) = Sum_{primes p} p^2 * (1-p-p^4) * log(p) / ((p^2-1)^2 * (p^2+p-1)^2) = -0.486606220169261905698805096547122238460686354267440350206456696497...

and gamma is the Euler-Mascheroni constant A001620. (End)