A088842 Denominator of the quotient sigma(7n)/sigma(n).

1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 57, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 57, 1, 1, 1, 1, 1, 1, 8

Offset: 1

Author

Labos Elemer, Nov 04 2003

Keywords

• nonn
• mult
• frac

Comments

Sum of powers of 7 dividing n. - Amiram Eldar, Nov 27 2022

Links

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

Crossrefs

Cf. A000203 (sigma), A001620, A088841 (numerators), A283078 (sigma(7n)).

Cf. A080278, A038712, A088837, A088838, A088839, A088840.

Programs

Mathematica

Table[Denominator[DivisorSigma[1, 7*n]/DivisorSigma[1, n]], {n, 1, 128}] (* corrected by Ilya Gutkovskiy, Dec 15 2020 *)
a[n_] := (7^(IntegerExponent[n, 7] + 1) - 1)/6; Array[a, 100] (* Amiram Eldar, Nov 27 2022 *)

PARI

a(n) = denominator(sigma(7*n)/sigma(n)); \\ Michel Marcus, Dec 15 2020

PARI

a(n) = (7^(valuation(n, 7) + 1) - 1)/6; \\ Amiram Eldar, Nov 27 2022

Formula

G.f.: Sum_{k>=0} 7^k * x^(7^k) / (1 - x^(7^k)). - Ilya Gutkovskiy, Dec 15 2020

From Amiram Eldar, Nov 27 2022: (Start)

Multiplicative with a(7^e) = (7^(e+1)-1)/6, and a(p^e) = 1 for p != 7.

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

Sum_{k=1..n} a(k) ~ n*log_7(n) + (1/2 + (gamma - 1)/log(7))*n, where gamma is Euler's constant (A001620). (End)