A321821 a(n) = Sum_{d|n, d==1 mod 4} d^5 - Sum_{d|n, d==3 mod 4} d^5.

1, 1, -242, 1, 3126, -242, -16806, 1, 58807, 3126, -161050, -242, 371294, -16806, -756492, 1, 1419858, 58807, -2476098, 3126, 4067052, -161050, -6436342, -242, 9768751, 371294, -14290100, -16806, 20511150, -756492, -28629150, 1, 38974100

Offset: 1

Author

N. J. A. Sloane, Nov 24 2018

Keywords

• sign
• easy
• 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

Column k=5 of A322143.

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

Cf. A000265.

Programs

Mathematica

s[n_, r_] := DivisorSum[n, #^5 &, Mod[#, 4] == r &]; a[n_] := s[n, 1] - s[n, 3]; Array[a, 30] (* Amiram Eldar, Nov 26 2018 *)
Table[With[{d=Divisors[n]},Total[Select[d,Mod[#,4]==1&]^5]-Total[Select[ d,Mod[ #,4]==3&]^5]],{n,40}] (* Harvey P. Dale, Dec 15 2018 *)
f[p_, e_] := If[Mod[p, 4] == 1, ((p^5)^(e+1)-1)/(p^5-1), ((-p^5)^(e+1)-1)/(-p^5-1)]; f[2, e_] := 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 60] (* Amiram Eldar, Sep 27 2023 *)

PARI

apply( A321821(n)=sumdiv(n>>valuation(n,2),d,(2-d%4)*d^5), [1..40]) \\ M. F. Hasler, Nov 26 2018

Formula

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

G.f.: Sum_{k>=1} (-1)^(k-1)*(2*k - 1)^5*x^(2*k-1)/(1 - x^(2*k-1)). - Ilya Gutkovskiy, Dec 22 2018

Multiplicative with a(2^e) = 1, and for an odd prime p, ((p^5)^(e+1)-1)/(p^5-1) if p == 1 (mod 4) and ((-p^5)^(e+1)-1)/(-p^5-1) if p == 3 (mod 4). - Amiram Eldar, Sep 27 2023

a(n) = Sum_{d|n} d^5*sin(d*Pi/2). - Ridouane Oudra, Jun 23 2024