A321835 a(n) = Sum_{d|n, n/d==1 mod 4} d^11 - Sum_{d|n, n/d==3 mod 4} d^11.

1, 2048, 177146, 4194304, 48828126, 362795008, 1977326742, 8589934592, 31380882463, 100000002048, 285311670610, 743004176384, 1792160394038, 4049565167616, 8649707208396, 17592186044416, 34271896307634, 64268047284224, 116490258898218

Offset: 1

Author

N. J. A. Sloane, Nov 24 2018

Keywords

• nonn
• 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

Cf. A101455.

Cf. A321807 - A321836 for related sequences.

Glaisher's E'_i (i=0..12): A002654, A050469, A050470, A050471, A050468, A321829, A321830, A321831, A321832, A321833, A321834, this sequence, A321836.

Programs

Mathematica

s[n_,r_] := DivisorSum[n, #^11 &, Mod[n/#,4]==r &]; a[n_] := s[n,1] - s[n,3]; Array[a, 30] (* Amiram Eldar, Nov 26 2018 *)
s[n_] := If[OddQ[n], (-1)^((n-1)/2), 0]; (* A101455 *)
f[p_, e_] := (p^(11*e+11) - s[p]^(e+1))/(p^11 - s[p]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 04 2023 *)

PARI

apply( a(n)=sumdiv(n,d,if(bittest(n\d,0),(2-n\d%4)*d^11)), [1..30]) \\ M. F. Hasler, Nov 26 2018

Formula

G.f.: Sum_{k>=1} k^11*x^k/(1 + x^(2*k)). - Ilya Gutkovskiy, Nov 26 2018

From Amiram Eldar, Nov 04 2023: (Start)

Multiplicative with a(p^e) = (p^(11*e+11) - A101455(p)^(e+1))/(p^11 - A101455(p)).

Sum_{k=1..n} a(k) ~ c * n^12 / 12, where c = beta(12) = 0.99999812235..., and beta is the Dirichlet beta function. (End)

a(n) = Sum_{d|n} (n/d)^11*sin(d*Pi/2). - Ridouane Oudra, Sep 27 2024