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

Author

Ilya Gutkovskiy, Oct 17 2019

Keywords

• nonn
• mult
• easy

Comments

Inverse Moebius transform of A003557.

Dirichlet convolution of A000203 with A097945.

Links

• 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.

Crossrefs

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

Programs

Mathematica

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 *)

PARI

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

Formula

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)