A321561 a(n) = Sum_{d divides n} (-1)^(d + n/d) * d^5.

1, -33, 244, -993, 3126, -8052, 16808, -31713, 59293, -103158, 161052, -242292, 371294, -554664, 762744, -1014753, 1419858, -1956669, 2476100, -3104118, 4101152, -5314716, 6436344, -7737972, 9768751, -12252702, 14408200, -16690344, 20511150

Offset

1,2

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

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^5*x^k/(1 + x^k). - Ilya Gutkovskiy, Nov 27 2018

Multiplicative with a(2^e) = -3*(5*2^(5*e+1) + 21)/31, and a(p^e) = (p^(5*e+5) - 1)/(p^5 - 1) for p > 2. - Amiram Eldar, Nov 22 2022

Mathematica

a[n_] := DivisorSum[n, (-1)^(# + n/#)*#^5 &]; Array[a, 50] (* Amiram Eldar, Nov 27 2018 *)

Prog

(PARI) apply( A321561(n)=sumdiv(n, d, (-1)^(n\d-d)*d^5), [1..30]) \\ M. F. Hasler, Nov 26 2018
(Magma) m:=50; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&+[(-1)^(k+1)*k^5*x^k/(1 + x^k) : k in [1..2*m]]) )); // G. C. Greubel, Nov 28 2018
(Sage) s=(sum((-1)^(k+1)*k^5*x^k/(1 + x^k)  for k in (1..50))).series(x, 50); a = s.coefficients(x, sparse=False); a[1:] # G. C. Greubel, Nov 28 2018

Crossrefs

Column k=5 of A322083.

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

Sequence in context: A178448 A351268 A088703 * A034679 A351300 A017673

Adjacent sequences: A321558 A321559 A321560 * A321562 A321563 A321564

Keyword

sign,mult

Author

N. J. A. Sloane, Nov 23 2018

Status

approved