A349694 Dirichlet convolution of the squarefree kernel function (A007947) with itself.

1, 4, 6, 8, 10, 24, 14, 12, 15, 40, 22, 48, 26, 56, 60, 16, 34, 60, 38, 80, 84, 88, 46, 72, 35, 104, 24, 112, 58, 240, 62, 20, 132, 136, 140, 120, 74, 152, 156, 120, 82, 336, 86, 176, 150, 184, 94, 96, 63, 140, 204, 208, 106, 96, 220, 168, 228, 232, 118, 480

Offset

1,2

Links

Table of n, a(n) for n=1..60.

Formula

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

a(n) = Sum_{d|n} A007947(d) * A007947(n/d).

a(n) = Sum_{d|n} abs(A097945(d)) * A191750(n/d).

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

From Vaclav Kotesovec, Nov 26 2021: (Start)

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

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

Sum_{k=1..n} a(k) ~ Pi^2 * f(2)^2 * n^2 / 144 * (Pi^2 * (2*log(n) + 4*gamma - 1 + 4*f'(2)/f(2)) + 24*zeta'(2)), where f(2) = Product_{primes p} (1 - 2/p^2 + 1/p^3) = A065464 = 0.428249505677094440218765707581823546121298513355936144..., f'(2) = f(2) * Sum_{primes p} log(p) * (3*p - 2) / (p^3 - 2*p + 1) = 0.6293283828324697510445630056425352981207558777167836747744750359407300858..., zeta'(2) = -A073002 and gamma is the Euler-Mascheroni constant A001620. (End)

Mathematica

Table[Sum[Last[Select[Divisors[d], SquareFreeQ]] Last[Select[Divisors[n/d], SquareFreeQ]], {d, Divisors[n]}], {n, 1, 60}]
f[p_, e_] := (e - 1)*p^2 + 2*p; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 60] (* Amiram Eldar, Nov 25 2021 *)

Prog

(PARI)
A007947(n) = factorback(factorint(n)[, 1]); \\ From A007947
A349694(n) = sumdiv(n, d, A007947(n/d)*A007947(d)); \\ Antti Karttunen, Nov 25 2021

Crossrefs

Cf. A007947, A097945, A176345, A191750.

Sequence in context: A050835 A369139 A294243 * A298473 A054284 A295287

Adjacent sequences: A349691 A349692 A349693 * A349695 A349696 A349697

Keyword

nonn,mult

Author

Ilya Gutkovskiy, Nov 25 2021

Status

approved