A158522 -id:A158522 - cp's OEIS frontend

A365498 Dirichlet g.f.: zeta(s) * Product_{p prime} (1 + 1/p^s - 1/p^(3*s)).

1, 2, 2, 2, 2, 4, 2, 1, 2, 4, 2, 4, 2, 4, 4, 1, 2, 4, 2, 4, 4, 4, 2, 2, 2, 4, 1, 4, 2, 8, 2, 1, 4, 4, 4, 4, 2, 4, 4, 2, 2, 8, 2, 4, 4, 4, 2, 2, 2, 4, 4, 4, 2, 2, 4, 2, 4, 4, 2, 8, 2, 4, 4, 1, 4, 8, 2, 4, 4, 8, 2, 2, 2, 4, 4, 4, 4, 8, 2, 2, 1, 4, 2, 8, 4, 4, 4

Offset: 1

Vaclav Kotesovec, Sep 06 2023

The number of unitary divisors of n that are cubefree numbers (A004709). - Amiram Eldar, Sep 06 2023

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000
Jon Maiga, Computer-generated formulas for A365498, Sequence Machine.

Cf. A001620, A004709, A056671, A365488, A365499.

Cf. A034444, A158522, A190867, A286324, A307428, A368172, A368248, A368885, A369310.

f[p_, e_] := If[e <= 2, 2, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 06 2023 *)

for(n=1, 100, print1(direuler(p=2, n, 1/(1-X) * (1 + X - X^3))[n], ", "))

Dirichlet g.f.: zeta(s)^2 * Product_{p prime} (1 - 1/p^(2*s) - 1/p^(3*s) + 1/p^(4*s)).
Let f(s) = Product_{p prime} (1 - 1/p^(2*s) - 1/p^(3*s) + 1/p^(4*s)).
Sum_{k=1..n} a(k) ~ f(1) * n * (log(n) + 2*gamma - 1 + f'(1)/f(1)), where
f(1) = Product_{p prime} (1 - 1/p^2 - 1/p^3 + 1/p^4) = 0.5358961538283379998085026313185459506482223745141452711510108346133288...,
f'(1) = f(1) * Sum_{p prime} (-4 + 3*p + 2*p^2) * log(p) / (1 - p - p^2 + p^4) = f(1) * 1.4525924794451595590371439593828547341482465114411929136723476679...
and gamma is the Euler-Mascheroni constant A001620.
Multiplicative with a(p^e) = 2 if e <= 2, and 1 otherwise. - Amiram Eldar, Sep 06 2023
From Vaclav Kotesovec, Jan 27 2025: (Start)
Following formulas have been conjectured for this sequence by Sequence Machine, with each one giving the first 1000000 terms correctly:
a(n) = A056671(n) * A368885(n).
a(n) = A034444(n) / A368248(n).
a(n) = A158522(n) / A307428(n).
a(n) = A369310(n) / A190867(n).
a(n) = A286324(n) / A368172(n). (End)

A326415 Dirichlet g.f.: zeta(2*s) / zeta(s)^3.

Original entry on oeis.org

1, -3, -3, 4, -3, 9, -3, -4, 4, 9, -3, -12, -3, 9, 9, 4, -3, -12, -3, -12, 9, 9, -3, 12, 4, 9, -4, -12, -3, -27, -3, -4, 9, 9, 9, 16, -3, 9, 9, 12, -3, -27, -3, -12, -12, 9, -3, -12, 4, -12, 9, -12, -3, 12, 9, 12, 9, 9, -3, 36, -3, 9, -12, 4, 9, -27, -3, -12, 9, -27, -3, -16, -3, 9, -12

Offset: 1

Ilya Gutkovskiy, Oct 18 2019

Moebius transform applied twice to A008836.

Dirichlet inverse of A048691.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Ilya Gutkovskiy, Scatter plot of partial sums of A326415 up to n=10000.

Cf. A000005, A001221, A001222, A007427, A007428, A008836, A010052, A034444, A046951, A048691, A158522.

Table[Sum[MoebiusMu[n/d] (-1)^PrimeOmega[d] 2^PrimeNu[d], {d, Divisors[n]}], {n, 1, 75}]
a[n_] := If[n == 1, n, -Sum[If[d < n, DivisorSigma[0, (n/d)^2] a[d], 0], {d, Divisors[n]}]]; Table[a[n], {n, 1, 75}]
f[p_, e_] := If[e == 1, -3, (-1)^e*4]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Oct 26 2020 *)

a(n) = Sum_{d|n} mu(n/d) * (-1)^bigomega(d) * 2^omega(d), where mu = A008683, bigomega = A001222 and omega = A001221.
a(1) = 1; a(n) = -Sum_{d|n, dA000005.
a(n) = Sum_{d|n} A008836(n/d) * A007427(d).
a(n) = Sum_{d|n} A010052(n/d) * A007428(d).
Multiplicative with a(p^e) = -3 if e = 1, and 4*(-1)^e otherwise. - Amiram Eldar, Oct 26 2020
b(n) = abs( a(n) ) is multiplicative with b(p) = 3 and b(p^e) = 4 for e > 1 and prime p. Its Dirichlet g.f. is: zeta(s)^3 / zeta(2*s)^2. - Werner Schulte, Jan 18 2023

cp's OEIS Frontend

A365498 Dirichlet g.f.: zeta(s) * Product_{p prime} (1 + 1/p^s - 1/p^(3*s)).

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