A321825 a(n) = Sum_{d|n, d==1 (mod 4)} d^9 - Sum_{d|n, d==3 (mod 4)} d^9.
1, 1, -19682, 1, 1953126, -19682, -40353606, 1, 387400807, 1953126, -2357947690, -19682, 10604499374, -40353606, -38441425932, 1, 118587876498, 387400807, -322687697778, 1953126, 794239673292, -2357947690, -1801152661462, -19682, 3814699218751
Offset: 1
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
Programs
-
Mathematica
s[n_, r_] := DivisorSum[n, #^9 &, 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^9)^(e+1)-1)/(p^9-1), ((-p^9)^(e+1)-1)/(-p^9-1)]; f[2, e_] := 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 60] (* Amiram Eldar, Sep 27 2023 *)
-
PARI
apply( A321825(n)=sumdiv(n>>valuation(n,2),d,(2-d%4)*d^9), [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)^9*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^9)^(e+1)-1)/(p^9-1) if p == 1 (mod 4) and ((-p^9)^(e+1)-1)/(-p^9-1) if p == 3 (mod 4). - Amiram Eldar, Sep 27 2023
a(n) = Sum_{d|n} d^9*sin(d*Pi/2). - Ridouane Oudra, Aug 18 2024