A335032 Dirichlet g.f.: zeta(s) * zeta(s-1) * Product_{p prime} (1 + p^(1 - s) - p^(-s)).

1, 4, 6, 10, 10, 24, 14, 22, 21, 40, 22, 60, 26, 56, 60, 46, 34, 84, 38, 100, 84, 88, 46, 132, 55, 104, 66, 140, 58, 240, 62, 94, 132, 136, 140, 210, 74, 152, 156, 220, 82, 336, 86, 220, 210, 184, 94, 276, 105, 220, 204, 260, 106, 264, 220, 308, 228, 232, 118

Offset

1,2

Comments

Dirichlet convolution of A000203 with abs(A097945).

Links

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Vaclav Kotesovec, Graph - the asymptotic ratio (1000000 terms).

Formula

Dirichlet g.f.: zeta(s) * zeta(s-1)^2 / zeta(2*s-2) * Product_{primes p} (1 - 1/(p^s + p)).

Dirichlet g.f.: zeta(s) * zeta(s-1)^2 * Product_{primes p} (1 + p^(1-2*s) - p^(2-2*s) - p^(-s)).

Let f(s) = Product_{primes p} (1 - 1/(p^s + p)), then Sum_{k=1..n} a(k) ~ n^2 * ((log(n)/2 + gamma - 3*zeta'(2)/Pi^2 - 1/4)*f(2) + f'(2)/2), where f(2) = A065463 = Product_{primes p} (1 - 1/(p*(p+1))) = 0.7044422009991655927366033503266372..., f'(2) = f(2) * Sum_{primes p} p*log(p) / ((p+1)*(p^2+p-1)) = 0.23219454323726621271960146689644280341444084188447499043209938838191022838..., for zeta'(2) see A073002 and gamma is the Euler-Mascheroni constant A001620.

a(n) = Sum_{d|n} A176345(d). - Ridouane Oudra, Jan 14 2022

Multiplicative with a(p^e) = sigma(p^e) + p^e - 1. - Amiram Eldar, Dec 25 2022

Mathematica

Table[Sum[DivisorSigma[1, n/d] * Abs[MoebiusMu[d]] * EulerPhi[d], {d, Divisors[n]}], {n, 1, 100}]

Prog

(PARI) for(n=1, 100, print1(direuler(p=2, n, (1 + p*X - X)/(1 - X)/(1 - p*X))[n], ", "))
(PARI) for(n=1, 100, print1(direuler(p=2, n, (1 + p*X/(1 - X))/(1 - p*X))[n], ", "))

Crossrefs

Cf. A000203, A001620, A097945, A176345, A326306.

Sequence in context: A088682 A102415 A117003 * A349132 A057670 A171615

Adjacent sequences: A335029 A335030 A335031 * A335033 A335034 A335035

Keyword

nonn,mult

Author

Vaclav Kotesovec, Jun 20 2020

Status

approved