A191750 -id:A191750 - cp's OEIS frontend

A326306 Dirichlet g.f.: zeta(s) * zeta(s-1) * Product_{p prime} (1 - p^(1 - s) + p^(-s)).

1, 2, 2, 4, 2, 4, 2, 8, 5, 4, 2, 8, 2, 4, 4, 16, 2, 10, 2, 8, 4, 4, 2, 16, 7, 4, 14, 8, 2, 8, 2, 32, 4, 4, 4, 20, 2, 4, 4, 16, 2, 8, 2, 8, 10, 4, 2, 32, 9, 14, 4, 8, 2, 28, 4, 16, 4, 4, 2, 16, 2, 4, 10, 64, 4, 8, 2, 8, 4, 8, 2, 40, 2, 4, 14, 8, 4, 8, 2, 32, 41, 4, 2, 16, 4

Offset: 1

Ilya Gutkovskiy, Oct 17 2019

Inverse Moebius transform of A003557.

Dirichlet convolution of A000203 with A097945.

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000
Steven R. Finch, Idempotents and Nilpotents Modulo n, arXiv:math/0605019 [math.NT], 2006-2017.

Cf. A000010, A000079 (fixed points), A000203, A003557, A007947, A008683, A098108 (parity of a(n)), A191750, A300717, A335032.

Table[Sum[d/Last[Select[Divisors[d], SquareFreeQ]], {d, Divisors[n]}], {n, 1, 85}]
Table[Sum[MoebiusMu[n/d] EulerPhi[n/d] DivisorSigma[1, d], {d, Divisors[n]}], {n, 1, 85}]
f[p_, e_] := 1 + (p^e-1)/(p-1); a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Oct 14 2020 *)

for(n=1, 100, print1(direuler(p=2, n, (1 - p*X + X)/(1 - X)/(1 - p*X))[n], ", ")) \\ Vaclav Kotesovec, Jun 14 2020

G.f.: Sum_{k>=1} (k / rad(k)) * x^k / (1 - x^k), where rad = A007947.
a(n) = Sum_{d|n} A003557(d).
a(n) = Sum_{d|n} mu(n/d) * phi(n/d) * sigma(d), where mu = A008683, phi = A000010 and sigma = A000203.
a(p) = 2, where p is prime.
From Vaclav Kotesovec, Jun 20 2020: (Start)
Dirichlet g.f.: zeta(s) * Product_{primes p} (1 + 1/(p^s - p)).
Dirichlet g.f.: zeta(s) * zeta(2*s-2) * Product_{primes p} (1 + p^(1-2*s) - p^(2-2*s) + p^(-s)). (End)
Multiplicative with a(p^e) = 1 + (p^e-1)/(p-1). - Amiram Eldar, Oct 14 2020
From Richard L. Ollerton, May 07 2021: (Start)
a(n) = Sum_{k=1..n} mu(n/gcd(n,k))*sigma(gcd(n,k)).
a(n) = Sum_{k=1..n} mu(gcd(n,k))*sigma(n/gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)). (End)

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

Original entry on oeis.org

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

Ilya Gutkovskiy, Nov 25 2021

Cf. A007947, A097945, A176345, A191750.

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 *)

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

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)

cp's OEIS Frontend

A326306 Dirichlet g.f.: zeta(s) * zeta(s-1) * Product_{p prime} (1 - p^(1 - s) + p^(-s)).

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