A321814 Sum of 10th powers of odd divisors of n.

1, 1, 59050, 1, 9765626, 59050, 282475250, 1, 3486843451, 9765626, 25937424602, 59050, 137858491850, 282475250, 576660215300, 1, 2015993900450, 3486843451, 6131066257802, 9765626, 16680163512500, 25937424602, 41426511213650, 59050

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) = A013958(A000265(n)) = sigma_10(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)^10*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^(10*e+10)-1)/(p^10-1) for p > 2.

Sum_{k=1..n} a(k) ~ c * n^11, where c = zeta(11)/22 = 0.045477... . (End)

Mathematica

a[n_] := DivisorSum[n, #^10 &, OddQ[#] &]; Array[a, 20] (* Amiram Eldar, Dec 07 2018 *)

Prog

(PARI) apply( A321814(n)=sigma(n>>valuation(n, 2), 10), [1..30]) \\ M. F. Hasler, Nov 26 2018
(Python)
from sympy import divisor_sigma
def A321814(n): return int(divisor_sigma(n>>(~n&n-1).bit_length(), 10)) # Chai Wah Wu, Jul 16 2022

Crossrefs

Column k=10 of A285425.

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

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

Cf. A000265, A013669, A013958.

Sequence in context: A017434 A017566 A352038 * A288886 A210172 A333056

Adjacent sequences: A321811 A321812 A321813 * A321815 A321816 A321817

Keyword

nonn,mult

Author

N. J. A. Sloane, Nov 24 2018

Status

approved