A321551 a(n) = Sum_{d|n} (-1)^(d-1)*d^12.

1, -4095, 531442, -16781311, 244140626, -2176254990, 13841287202, -68736258047, 282430067923, -999755863470, 3138428376722, -8918293480462, 23298085122482, -56680071092190, 129746582562692, -281543712968703, 582622237229762, -1156551128144685, 2213314919066162, -4096999772640686

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^(12*e + 12) - 1)/(p^12 - 1); f[2, e_] := 2 - (2^(12*e + 12) - 1)/4095; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 20] (* Amiram Eldar, Nov 04 2022 *)

PARI

apply( a(n)=sumdiv(n, d, (-1)^(d-1)*d^12), [1..30]) \\ M. F. Hasler, Nov 26 2018

Formula

G.f.: Sum_{k>=1} (-1)^(k-1)*k^12*x^k/(1 - x^k). - Ilya Gutkovskiy, Dec 24 2018

Multiplicative with a(2^e) = 2 - (2^(12*e + 12) - 1)/4095, and a(p^e) = (p^(12*e + 12) - 1)/(p^12 - 1) for p > 2. - Amiram Eldar, Nov 04 2022