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

Author

Ilya Gutkovskiy, Nov 25 2021

Keywords

• nonn
• mult

Crossrefs

Cf. A007947, A097945, A176345, A191750.

Programs

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 *)

PARI

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

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)