A321562 a(n) = Sum_{d divides n} (-1)^(d + n/d) * d^6.

1, -65, 730, -4033, 15626, -47450, 117650, -257985, 532171, -1015690, 1771562, -2944090, 4826810, -7647250, 11406980, -16510913, 24137570, -34591115, 47045882, -63019658, 85884500, -115151530, 148035890, -188329050, 244156251, -313742650

Offset: 1

Author

N. J. A. Sloane, Nov 23 2018

Keywords

• sign
• mult

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.

Crossrefs

Column k=6 of A322083.

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

Programs

Magma

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&+[(-1)^(k+1)*k^6*x^k/(1 + x^k) : k in [1..2*m]]) )); // G. C. Greubel, Nov 28 2018

Mathematica

a[n_] := DivisorSum[n, (-1)^(# + n/#)*#^6 &]; Array[a, 50] (* Amiram Eldar, Nov 27 2018 *)

PARI

apply( A321562(n)=sumdiv(n, d, (-1)^(n\d-d)*d^6), [1..30]) \\ M. F. Hasler, Nov 26 2018

Sage

s=(sum((-1)^(k+1)*k^6*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

Formula

G.f.: Sum_{k>=1} (-1)^(k+1)*k^6*x^k/(1 + x^k). - Ilya Gutkovskiy, Nov 27 2018

Multiplicative with a(2^e) = -(31*2^(6*e+1) + 127)/63, and a(p^e) = (p^(6*e+6) - 1)/(p^6 - 1) for p > 2. - Amiram Eldar, Nov 22 2022