A373438 Expansion of Sum_{k>=1} k * x^(3^(k-1)) / (1 - x^(3^(k-1))).

1, 1, 3, 1, 1, 3, 1, 1, 6, 1, 1, 3, 1, 1, 3, 1, 1, 6, 1, 1, 3, 1, 1, 3, 1, 1, 10, 1, 1, 3, 1, 1, 3, 1, 1, 6, 1, 1, 3, 1, 1, 3, 1, 1, 6, 1, 1, 3, 1, 1, 3, 1, 1, 10, 1, 1, 3, 1, 1, 3, 1, 1, 6, 1, 1, 3, 1, 1, 3, 1, 1, 6, 1, 1, 3, 1, 1, 3, 1, 1, 15, 1, 1, 3, 1, 1, 3, 1, 1, 6, 1, 1, 3, 1, 1, 3, 1, 1, 6, 1, 1, 3, 1, 1, 3

Offset: 1

Author

Ilya Gutkovskiy, Jun 05 2024

Keywords

• nonn
• mult
• easy

Links

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

Crossrefs

Cf. A000217, A051064, A080278, A115364.

Programs

Mathematica

nmax = 105; CoefficientList[Series[Sum[k x^(3^(k - 1))/(1 - x^(3^(k - 1))), {k, 1, Floor[Log[3, nmax]] + 1}], {x, 0, nmax}], x] // Rest
Table[Binomial[IntegerExponent[3 n, 3] + 1, 2], {n, 1, 105}]

PARI

a(n) = {my(e = valuation(n, 3)); (e+1)*(e+2)/2;} \\ Amiram Eldar, Jun 27 2024

Formula

a(n) = A000217(A051064(n)).

From Vaclav Kotesovec, Jun 25 2024: (Start)

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

Sum_{k=1..n} a(k) ~ 9*n/4 - log(n)*(log(n) + 2*log(6*Pi))/(4*log(3)^2). (End)

Multiplicative with a(p^e) = (e+1)*(e+2)/2 if p = 3 and 1 if p != 3. - Amiram Eldar, Jun 27 2024