A373217 Expansion of Sum_{k>=0} x^(7^k) / (1 - x^(7^k)).

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

Offset: 1

Author

Seiichi Manyama, May 28 2024

Keywords

• nonn
• mult
• easy

Comments

The number of powers of 7 that divide n. - Amiram Eldar, Mar 29 2025

Links

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

Crossrefs

Cf. A001511, A051064, A055457, A115362, A373216.

Cf. A000005, A008683, A214411, A373221.

Programs

Mathematica

a[n_] := 1 + IntegerExponent[n, 7]; Array[a, 100] (* Amiram Eldar, May 29 2024 *)

PARI

a(n) = valuation(n, 7)+1;

Formula

G.f. A(x) satisfies A(x) = x/(1 - x) + A(x^7).

a(7*n+1) = a(7*n+2) = ... = (7*n+6) = 1 and a(7*n+7) = 1 + a(n+1) for n >= 0.

Multiplicative with a(p^e) = e+1 if p = 7, 1 otherwise.

a(n) = -Sum_{d|n} mu(7*d) * tau(n/d).

a(n) = A214411(n) + 1.

From Amiram Eldar, May 29 2024: (Start)

Dirichlet g.f.: (7^s/(7^s-1)) * zeta(s).

Sum_{k=1..n} a(k) ~ (7/6) * n. (End)

G.f.: Sum_{i>=1, j>=0} x^(i*7^j). - Seiichi Manyama, Mar 23 2025

a(n) = A214411(7*n). - R. J. Mathar, Jun 28 2025