A328484 Dirichlet g.f.: zeta(s)^2 / (1 - 3^(-s)).

1, 2, 3, 3, 2, 6, 2, 4, 6, 4, 2, 9, 2, 4, 6, 5, 2, 12, 2, 6, 6, 4, 2, 12, 3, 4, 10, 6, 2, 12, 2, 6, 6, 4, 4, 18, 2, 4, 6, 8, 2, 12, 2, 6, 12, 4, 2, 15, 3, 6, 6, 6, 2, 20, 4, 8, 6, 4, 2, 18, 2, 4, 12, 7, 4, 12, 2, 6, 6, 8, 2, 24, 2, 4, 9, 6, 4, 12, 2, 10, 15, 4, 2, 18, 4

Offset: 1

Author

Ilya Gutkovskiy, Oct 16 2019

Keywords

• nonn
• mult

Comments

Inverse Moebius transform of A051064.

Dirichlet convolution of A000005 with characteristic function of powers of 3.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000005, A051064, A129628.

Cf. A007949, A372713, A373438, A035191.

Programs

Maple

seq(add(padic[ordp](3*d, 3), d in numtheory[divisors](n)), n=1..100); # Ridouane Oudra, Sep 30 2024

Mathematica

Table[DivisorSum[n, IntegerExponent[3 #, 3] &], {n, 1, 85}]
nmax = 85; CoefficientList[Series[Sum[Sum[x^(i 3^j)/(1 - x^(i 3^j)), {j, 0, Floor[Log[3, nmax]] + 1}], {i, 1, nmax}], {x, 0, nmax}], x] // Rest
f[p_, e_] := If[p == 3, (e + 1)*(e + 2)/2, e + 1]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Dec 02 2020 *)

Formula

G.f.: Sum_{i>=1} Sum_{j>=0} x^(i*3^j) / (1 - x^(i*3^j)).

a(n) = Sum_{d|n} A051064(d).

Sum_{k=1..n} a(k) ~ 3*n*(log(n)/2 - log(3)/4 - 1/2 + gamma), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Oct 17 2019

Multiplicative with a(p^e) = (e+1)*(e+2)/2 if p=3, and e+1 otherwise. - Amiram Eldar, Dec 02 2020

From Ridouane Oudra, Sep 30 2024: (Start)

a(n) = Sum_{i=0..A007949(n)} tau(n/3^i).

a(n) = Sum_{d|3*n} A007949(d).

a(n) = (1/2)*A051064(n)*A372713(n).

a(n) = (1/2)*(A051064(n) + 1)*A000005(n).

a(n) = A373438(n)*A035191(n). (End)