A373216 Expansion of Sum_{k>=0} x^(6^k) / (1 - x^(6^k)).

1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3

Offset

1,6

Links

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Formula

G.f. A(x) satisfies A(x) = x/(1 - x) + A(x^6).

a(6*n+1) = a(6*n+2) = ... = (6*n+5) = 1 and a(6*n+6) = 1 + a(n+1) for n >= 0.

a(n) = A122841(n) + 1.

G.f.: Sum_{i>=1, j>=0} x^(i*6^j). - Seiichi Manyama, Mar 23 2025

a(n) = A122841(6*n). - R. J. Mathar, Jun 28 2025

From Amiram Eldar, Feb 19 2026: (Start)

Dirichlet g.f.: (6^s/(6^s-1)) * zeta(s).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 6/5. (End)

Sum_{k=1..n} a(k) ~ 6*n/5 - log(n)/log(36). - Vaclav Kotesovec, Feb 20 2026

Maple

a:= n-> padic[ordp](n, 6)+1:
seq(a(n), n=1..108);  # Alois P. Heinz, Feb 20 2026

Mathematica

a[n_] := IntegerExponent[n, 6] + 1; Array[a, 108] (* Amiram Eldar, Feb 19 2026 *)

Prog

(PARI) a(n) = valuation(n, 6)+1;

Crossrefs

Cf. A001511, A051064, A055457, A115362, A373217.

Cf. A122841, A373220.

Sequence in context: A242073 A245477 A319689 * A303824 A106751 A325469

Adjacent sequences: A373213 A373214 A373215 * A373217 A373218 A373219

Keyword

nonn,easy

Author

Seiichi Manyama, May 28 2024

Status

approved