A321815 Sum of 11th powers of odd divisors of n.

1, 1, 177148, 1, 48828126, 177148, 1977326744, 1, 31381236757, 48828126, 285311670612, 177148, 1792160394038, 1977326744, 8649804864648, 1, 34271896307634, 31381236757, 116490258898220, 48828126, 350279478046112, 285311670612, 952809757913928

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) = A013959(A000265(n)) = sigma_11(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)^11*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^(11*e+11)-1)/(p^11-1) for p > 2.

Sum_{k=1..n} a(k) ~ c * n^12, where c = zeta(12)/24 = 691*Pi^12/15324309000 = 0.0416769... . (End)

Mathematica

a[n_] := DivisorSum[n, #^11&, OddQ[#]&]; Array[a, 20] (* Amiram Eldar, Dec 07 2018 *)

Prog

(PARI) apply( A321815(n)=sigma(n>>valuation(n, 2), 11), [1..30]) \\ M. F. Hasler, Nov 26 2018
(GAP) List(List(List([1..25], j->DivisorsInt(j)), i->Filtered(i, k->IsOddInt(k))), m->Sum(m, n->n^11)); # Muniru A Asiru, Dec 07 2018
(Python)
from sympy import divisor_sigma
def A321815(n): return int(divisor_sigma(n>>(~n&n-1).bit_length(), 11)) # Chai Wah Wu, Jul 16 2022

Crossrefs

Column k=11 of A285425.

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

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

Cf. A000265, A013670, A013959.

Sequence in context: A017315 A017435 A017567 * A081867 A133972 A233480

Adjacent sequences: A321812 A321813 A321814 * A321816 A321817 A321818

Keyword

nonn,mult

Author

N. J. A. Sloane, Nov 24 2018

Status

approved