A321826 a(n) = Sum_{d|n, d==1 mod 4} d^10 - Sum_{d|n, d==3 mod 4} d^10.

1, 1, -59048, 1, 9765626, -59048, -282475248, 1, 3486725353, 9765626, -25937424600, -59048, 137858491850, -282475248, -576640684048, 1, 2015993900450, 3486725353, -6131066257800, 9765626, 16679598443904, -25937424600, -41426511213648, -59048, 95367441406251, 137858491850

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).

Index entries for sequences mentioned by Glaisher.

Formula

a(n) = a(A000265(n)). - M. F. Hasler, Nov 26 2018

G.f.: Sum_{k>=1} (-1)^(k-1)*(2*k - 1)^10*x^(2*k-1)/(1 - x^(2*k-1)). - Ilya Gutkovskiy, Dec 22 2018

Multiplicative with a(2^e) = 1, and for an odd prime p, ((p^10)^(e+1)-1)/(p^10-1) if p == 1 (mod 4) and ((-p^10)^(e+1)-1)/(-p^10-1) if p == 3 (mod 4). - Amiram Eldar, Sep 27 2023

a(n) = Sum_{d|n} d^10*sin(d*Pi/2). - Ridouane Oudra, Sep 04 2024

Mathematica

s[n_, r_] := DivisorSum[n, #^10 &, Mod[#, 4] == r &]; a[n_] := s[n, 1] - s[n, 3]; Array[a, 30] (* Amiram Eldar, Nov 26 2018 *)
f[p_, e_] := If[Mod[p, 4] == 1, ((p^10)^(e+1)-1)/(p^10-1), ((-p^10)^(e+1)-1)/(-p^10-1)]; f[2, e_] := 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 60] (* Amiram Eldar, Sep 27 2023 *)

Prog

(PARI) apply( A321826(n)=sumdiv(n>>valuation(n, 2), d, (2-d%4)*d^10), [1..40]) \\ M. F. Hasler, Nov 26 2018

Crossrefs

Column k=10 of A322143.

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

Cf. A000265.

Sequence in context: A235953 A343374 A359310 * A176763 A176769 A255893

Adjacent sequences: A321823 A321824 A321825 * A321827 A321828 A321829

Keyword

sign,easy,mult,changed

Author

N. J. A. Sloane, Nov 24 2018

Status

approved