A321818 a(n) = Sum_{d|n, n/d odd} d^8 for n > 0.

1, 256, 6562, 65536, 390626, 1679872, 5764802, 16777216, 43053283, 100000256, 214358882, 430047232, 815730722, 1475789312, 2563287812, 4294967296, 6975757442, 11021640448, 16983563042, 25600065536, 37828630724, 54875873792, 78310985282

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^8*x^k/(1 - x^(2*k)). - Ilya Gutkovskiy, Dec 22 2018

From Amiram Eldar, Nov 02 2022: (Start)

Multiplicative with a(2^e) = 2^(8*e) and a(p^e) = (p^(8*e+8)-1)/(p^8-1) for p > 2.

Sum_{k=1..n} a(k) ~ c * n^9, where c = 511*zeta(9)/4608 = 0.1111168... . (End)

Dirichlet g.f.: zeta(s)*zeta(s-8)*(1-1/2^s). - Amiram Eldar, Jan 09 2023

Mathematica

a[n_] := DivisorSum[n, #^8 &, OddQ[n/#] &]; Array[a, 24] (* Amiram Eldar, Nov 02 2022 *)

Prog

(PARI) apply( A321818(n)=sumdiv(n, d, if(bittest(n\d, 0), d^8)), [1..30]) \\ M. F. Hasler, Nov 26 2018

Crossrefs

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

Cf. A013667.

Sequence in context: A046457 A179645 A056585 * A231307 A206129 A236214

Adjacent sequences: A321815 A321816 A321817 * A321819 A321820 A321821

Keyword

nonn,mult

Author

N. J. A. Sloane, Nov 24 2018

Status

approved