A321549 a(n) = Sum_{d|n} (-1)^(d-1)*d^10.

1, -1023, 59050, -1049599, 9765626, -60408150, 282475250, -1074791423, 3486843451, -9990235398, 25937424602, -61978820950, 137858491850, -288972180750, 576660215300, -1100586419199, 2015993900450, -3567040850373, 6131066257802, -10249991283974, 16680163512500, -26533985367846, 41426511213650

Offset: 1

Author

N. J. A. Sloane, Nov 23 2018

Keywords

• sign
• mult

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.

Crossrefs

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

Programs

Mathematica

f[p_, e_] := (p^(10*e + 10) - 1)/(p^10 - 1); f[2, e_] := 2 - (2^(10*e + 10) - 1)/1023; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 24] (* Amiram Eldar, Nov 04 2022 *)

PARI

apply( a(n)=sumdiv(n, d, (-1)^(d-1)*d^10), [1..30]) \\ M. F. Hasler, Nov 26 2018

Formula

G.f.: Sum_{k>=1} (-1)^(k-1)*k^10*x^k/(1 - x^k). - Ilya Gutkovskiy, Dec 23 2018

Multiplicative with a(2^e) = 2 - (2^(10*e + 10) - 1)/1023, and a(p^e) = (p^(10*e + 10) - 1)/(p^10 - 1) for p > 2. - Amiram Eldar, Nov 04 2022