A321820 a(n) = Sum_{d|n, n/d odd} d^12 for n > 0.

1, 4096, 531442, 16777216, 244140626, 2176786432, 13841287202, 68719476736, 282430067923, 1000000004096, 3138428376722, 8916117225472, 23298085122482, 56693912379392, 129746582562692, 281474976710656, 582622237229762, 1156833558212608

Offset

1,2

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).

Index entries for sequences mentioned by Glaisher.

Formula

G.f.: Sum_{k>=1} k^12*x^k/(1 - x^(2*k)). - Ilya Gutkovskiy, Dec 22 2018

From Amiram Eldar, Nov 02 2022: (Start)

Multiplicative with a(2^e) = 2^(12*e) and a(p^e) = (p^(12*e+12)-1)/(p^12-1) for p > 2.

Sum_{k=1..n} a(k) ~ c * n^13, where c = 8191*zeta(13)/106496 = 0.0769231... . (End)

Dirichlet g.f.: zeta(s)*zeta(s-12)*(1-1/2^s). - Amiram Eldar, Jan 09 2023

Mathematica

a[n_] := DivisorSum[n, #^12 &, OddQ[n/#] &]; Array[a, 20] (* Amiram Eldar, Nov 02 2022 *)

Prog

(PARI) apply( A321820(n)=sumdiv(n, d, if(bittest(n\d, 0), d^12)), [1..30]) \\ M. F. Hasler, Nov 26 2018

Crossrefs

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

Cf. A013671.

Sequence in context: A017688 A008456 A030631 * A236218 A016962 A224395

Adjacent sequences: A321817 A321818 A321819 * A321821 A321822 A321823

Keyword

nonn,mult

Author

N. J. A. Sloane, Nov 24 2018

Status

approved