A321822 a(n) = Sum_{d|n, d==1 mod 4} d^6 - Sum_{d|n, d==3 mod 4} d^6.

1, 1, -728, 1, 15626, -728, -117648, 1, 530713, 15626, -1771560, -728, 4826810, -117648, -11375728, 1, 24137570, 530713, -47045880, 15626, 85647744, -1771560, -148035888, -728, 244156251, 4826810, -386889776, -117648, 594823322, -11375728

Offset

1,3

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

a(n) = a(A000265(n)). - M. F. Hasler, Nov 26 2018

G.f.: Sum_{k>=1} (-1)^(k-1)*(2*k - 1)^6*x^(2*k-1)/(1 - x^(2*k-1)). - Ilya Gutkovskiy, Dec 06 2018

Multiplicative with a(2^e) = 1, and for an odd prime p, ((p^6)^(e+1)-1)/(p^6-1) if p == 1 (mod 4) and ((-p^6)^(e+1)-1)/(-p^6-1) if p == 3 (mod 4). - Amiram Eldar, Sep 27 2023

a(n) = Sum_{d|n} d^6*sin(d*Pi/2). - Ridouane Oudra, Aug 17 2024

Mathematica

s[n_, r_] := DivisorSum[n, #^6 &, Mod[#, 4] == r &]; a[n_] := s[n, 1] - s[n, 3]; Array[a, 30] (* Amiram Eldar, Nov 26 2018 *)
f[p_, e_] := If[Mod[p, 4] == 1, ((p^6)^(e+1)-1)/(p^6-1), ((-p^6)^(e+1)-1)/(-p^6-1)]; f[2, e_] := 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 60] (* Amiram Eldar, Sep 27 2023 *)

Prog

(PARI) apply( A321822(n)=sumdiv(n>>valuation(n, 2), d, (2-d%4)*d^6), [1..40]) \\ M. F. Hasler, Nov 26 2018

Crossrefs

Column k=6 of A322143.

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

Cf. A000265.

Sequence in context: A178654 A094733 A378431 * A056084 A191345 A345744

Adjacent sequences: A321819 A321820 A321821 * A321823 A321824 A321825

Keyword

sign,easy,mult

Author

N. J. A. Sloane, Nov 24 2018

Status

approved