A321827 a(n) = Sum_{d|n, d==1 (mod 4)} d^11 - Sum_{d|n, d==3 (mod 4)} d^11.

1, 1, -177146, 1, 48828126, -177146, -1977326742, 1, 31380882463, 48828126, -285311670610, -177146, 1792160394038, -1977326742, -8649707208396, 1, 34271896307634, 31380882463, -116490258898218, 48828126, 350275523038332, -285311670610

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)^11*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^11)^(e+1)-1)/(p^11-1) if p == 1 (mod 4) and ((-p^11)^(e+1)-1)/(-p^11-1) if p == 3 (mod 4). - Amiram Eldar, Sep 27 2023

a(n) = Sum_{d|n} d^11*sin(d*Pi/2). - Ridouane Oudra, Sep 08 2024

Mathematica

s[n_, r_] := DivisorSum[n, #^11 &, 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^11)^(e+1)-1)/(p^11-1), ((-p^11)^(e+1)-1)/(-p^11-1)]; f[2, e_] := 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 60] (* Amiram Eldar, Sep 27 2023 *)

Prog

(PARI) apply( A321828(n)=sumdiv(n>>valuation(n, 2), d, (2-d%4)*d^11), [1..40]) \\ M. F. Hasler, Nov 26 2018

Crossrefs

Column k=11 of A322143.

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

Cf. A000265.

Sequence in context: A280447 A205177 A205276 * A016763 A016775 A016847

Adjacent sequences: A321824 A321825 A321826 * A321828 A321829 A321830

Keyword

sign,easy,mult

Author

N. J. A. Sloane, Nov 24 2018

Status

approved