A321813 Sum of 9th powers of odd divisors of n.

1, 1, 19684, 1, 1953126, 19684, 40353608, 1, 387440173, 1953126, 2357947692, 19684, 10604499374, 40353608, 38445332184, 1, 118587876498, 387440173, 322687697780, 1953126, 794320419872, 2357947692, 1801152661464, 19684, 3814699218751

Offset

1,3

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.

Formula

a(n) = A013957(A000265(n)) = sigma_9(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)^9*x^(2*k-1)/(1 - x^(2*k-1)). - Ilya Gutkovskiy, Dec 07 2018

From Amiram Eldar, Nov 02 2022: (Start)

Multiplicative with a(2^e) = 1 and a(p^e) = (p^(9*e+9)-1)/(p^9-1) for p > 2.

Sum_{k=1..n} a(k) ~ c * n^10, where c = zeta(10)/20 = Pi^10/1871100 = 0.0500497... . (End)

Mathematica

a[n_] := DivisorSum[n, #^9 &, OddQ[#] &]; Array[a, 20] (* Amiram Eldar, Dec 07 2018 *)

Prog

(PARI) apply( A321813(n)=sigma(n>>valuation(n, 2), 9), [1..30]) \\ M. F. Hasler, Nov 26 2018
(Python)
from sympy import divisor_sigma
def A321813(n): return int(divisor_sigma(n>>(~n&n-1).bit_length(), 9)) # Chai Wah Wu, Jul 16 2022

Crossrefs

Column k=9 of A285425.

Cf. A050999, A051000, A051001, A051002, A321810 - A321816 (analog for 2nd .. 12th powers).

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

Cf. A000265, A013668, A013957.

Sequence in context: A017433 A017565 A352037 * A081866 A288885 A253493

Adjacent sequences: A321810 A321811 A321812 * A321814 A321815 A321816

Keyword

nonn,mult

Author

N. J. A. Sloane, Nov 24 2018

Status

approved