A321810 Sum of 6th powers of odd divisors of n.

1, 1, 730, 1, 15626, 730, 117650, 1, 532171, 15626, 1771562, 730, 4826810, 117650, 11406980, 1, 24137570, 532171, 47045882, 15626, 85884500, 1771562, 148035890, 730, 244156251, 4826810, 387952660, 117650, 594823322, 11406980, 887503682

Offset: 1

Author

N. J. A. Sloane, Nov 24 2018

Keywords

• nonn
• mult

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..10000
• J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 4 and p. 8).
• Eric Weisstein's World of Mathematics, Odd Divisor Function.
• Index entries for sequences mentioned by Glaisher.

Crossrefs

Column k=6 of A285425.

Cf. A050999, A051000, A051001, A051002, A321811 - A321816 (analog for 2nd .. 12th powers).

Cf. A321543 - A321565, A321807 - A321836 for related sequences.

Cf. A000265, A013665, A013954.

Programs

Mathematica

f[2, e_] := 1; f[p_, e_] := (p^(6*e + 6) - 1)/(p^6 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 02 2022 *)

PARI

apply( A321810(n)=sigma(n>>valuation(n,2),6), [1..30]) \\ M. F. Hasler, Nov 26 2018

Python

from sympy import divisor_sigma
def A321810(n): return int(divisor_sigma(n>>(~n&n-1).bit_length(),6)) # Chai Wah Wu, Jul 16 2022

Formula

a(n) = A013954(A000265(n)) = sigma_6(odd part of n); in particular, a(2^k) = 1 for all k >= 0. - M. F. Hasler, Nov 26 2018

G.f.: Sum_{k>=1} (2*k - 1)^6*x^(2*k-1)/(1 - x^(2*k-1)). - Ilya Gutkovskiy, Dec 22 2018

From Amiram Eldar, Nov 02 2022: (Start)

Multiplicative with a(2^e) = 1 and a(p^e) = (p^(6*e+6)-1)/(p^6-1) for p > 2.

Sum_{k=1..n} a(k) ~ c * n^7, where c = zeta(7)/14 = 0.0720249... . (End)

a(n) + a(n/2)*2^6 = A013954(n) where a(.)=0 for non-integer arguments. - R. J. Mathar, Aug 15 2023