A349692 Dirichlet convolution of the gcd-sum function (A018804) with itself.

1, 6, 10, 25, 18, 60, 26, 88, 67, 108, 42, 250, 50, 156, 180, 280, 66, 402, 74, 450, 260, 252, 90, 880, 211, 300, 372, 650, 114, 1080, 122, 832, 420, 396, 468, 1675, 146, 444, 500, 1584, 162, 1560, 170, 1050, 1206, 540, 186, 2800, 435, 1266, 660, 1250, 210, 2232, 756

Offset

1,2

Links

Table of n, a(n) for n=1..55.

Vaclav Kotesovec, Asymptotics of Sum_{k=1..n} a(k) with a graph

Formula

Dirichlet g.f.: zeta(s-1)^4 / zeta(s)^2.

a(n) = Sum_{d|n} A018804(d) * A018804(n/d).

a(n) = Sum_{d|n} A029935(d) * A038040(n/d).

a(n) = Sum_{d|n} A344683(d) * A000203(n/d).

Multiplicative with a(p^e) = (e+1) * p^(e-2) * ((e+2)*(e+3)*p^2 - 2*e*(e+2)*p + e*(e-1))/6. - Amiram Eldar, Nov 25 2021

Mathematica

A018804[n_] := Sum[GCD[n, k], {k, 1, n}]; a[n_] := Sum[A018804[d] A018804[n/d], {d, Divisors[n]}]; Table[a[n], {n, 1, 55}]
f[p_, e_] := (e + 1)*p^(e - 2)*((e + 2)*(e + 3)*p^2 - 2*e*(e + 2)*p + e*(e - 1))/6; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 55] (* Amiram Eldar, Nov 25 2021 *)

Prog

(PARI)
A029935(n) = sumdiv(n, d, eulerphi(d)*eulerphi(n/d)); \\ From A029935.
A349692(n) = sumdiv(n, d, A029935(n/d)*d*numdiv(d)); \\ Antti Karttunen, Nov 25 2021

Crossrefs

Cf. A000203, A018804, A029935, A038040, A344683.

Sequence in context: A074289 A109099 A109095 * A351446 A323107 A077621

Adjacent sequences: A349689 A349690 A349691 * A349693 A349694 A349695

Keyword

nonn,mult

Author

Ilya Gutkovskiy, Nov 25 2021

Status

approved