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

1,6

Links

Antti Karttunen, Table of n, a(n) for n = 1..10000

Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)

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)

Prog

(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));

Crossrefs

Cf. A003415, A003557, A342001, A347962.

Cf. also A305809, A347963.

Sequence in context: A318366 A300252 A305796 * A020781 A327883 A007432

Adjacent sequences: A347958 A347959 A347960 * A347962 A347963 A347964

Keyword

nonn

Author

Antti Karttunen, Sep 24 2021

Status

approved