A321545 a(n) = Sum_{d|n} (-1)^(d-1)*d^6.

1, -63, 730, -4159, 15626, -45990, 117650, -266303, 532171, -984438, 1771562, -3036070, 4826810, -7411950, 11406980, -17043519, 24137570, -33526773, 47045882, -64988534, 85884500, -111608406, 148035890, -194401190, 244156251, -304089030, 387952660, -489306350, 594823322, -718639740, 887503682

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} (-1)^(k-1)*k^6*x^k/(1 - x^k). - Ilya Gutkovskiy, Dec 23 2018

Multiplicative with a(2^e) = 2 - (2^(6*e + 6) - 1)/63, and a(p^e) = (p^(6*e + 6) - 1)/(p^6 - 1) for p > 2. - Amiram Eldar, Nov 04 2022

Mathematica

f[p_, e_] := (p^(6*e + 6) - 1)/(p^6 - 1); f[2, e_] := 2 - (2^(6*e + 6) - 1)/63; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 30] (* Amiram Eldar, Nov 04 2022 *)

Prog

(PARI) apply( a(n)=sumdiv(n, d, (-1)^(d-1)*d^6), [1..30]) \\ M. F. Hasler, Nov 26 2018

Crossrefs

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

Sequence in context: A123866 A024004 A284927 * A201886 A232794 A091027

Adjacent sequences: A321542 A321543 A321544 * A321546 A321547 A321548

Keyword

sign,mult

Author

N. J. A. Sloane, Nov 23 2018

Status

approved