A326814 -id:A326814 - cp's OEIS frontend

A326815 Dirichlet g.f.: zeta(s)^3 * Product_{p prime} (1 - 2 * p^(-s)).

1, 1, 1, 0, 1, 1, 1, -2, 0, 1, 1, 0, 1, 1, 1, -5, 1, 0, 1, 0, 1, 1, 1, -2, 0, 1, -2, 0, 1, 1, 1, -9, 1, 1, 1, 0, 1, 1, 1, -2, 1, 1, 1, 0, 0, 1, 1, -5, 0, 0, 1, 0, 1, -2, 1, -2, 1, 1, 1, 0, 1, 1, 0, -14, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, -5, -5, 1, 1, 0, 1

Offset: 1

Ilya Gutkovskiy, Oct 19 2019

Inverse Moebius transform applied twice to A076479 (unitary Moebius function).

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000
Index entries for sequences computed from exponents in factorization of n

Cf. A000005, A001221, A005117 (positions of 1's), A007425, A008683, A038109 (positions of 0's), A046951, A076479, A080956, A326814.

Table[Sum[(-1)^PrimeNu[n/d] DivisorSigma[0, d], {d, Divisors[n]}], {n, 1, 85}]
f[p_, e_] := (e + 1)*(2 - e)/2; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Oct 26 2020 *)

A326815(n) = sumdiv(n,d,((-1)^omega(n/d))*numdiv(d)); \\ Antti Karttunen, Nov 17 2019

for(n=1, 100, print1(direuler(p=2, n, (1 - 2*X)/(1 - X)^3)[n], ", ")) \\ Vaclav Kotesovec, Aug 22 2021

a(n) = Sum_{d|n} (-1)^omega(n/d) * tau(d), where omega = A001221 and tau = A000005.
a(n) = Sum_{d|n} tau_3(n/d) * mu(d) * 2^omega(d), where tau_3 = A007425 and mu = A008683.
Multiplicative with a(p^e) = (e+1)*(2-e)/2 = A080956(e). - Amiram Eldar, Oct 26 2020

cp's OEIS Frontend

A326815 Dirichlet g.f.: zeta(s)^3 * Product_{p prime} (1 - 2 * p^(-s)).

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