A321823 a(n) = Sum_{d|n, d==1 mod 4} d^7 - Sum_{d|n, d==3 mod 4} d^7.
1, 1, -2186, 1, 78126, -2186, -823542, 1, 4780783, 78126, -19487170, -2186, 62748518, -823542, -170783436, 1, 410338674, 4780783, -893871738, 78126, 1800262812, -19487170, -3404825446, -2186, 6103593751, 62748518, -10455572420, -823542
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
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Mathematica
s[n_, r_] := DivisorSum[n, #^7 &, 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^7)^(e+1)-1)/(p^7-1), ((-p^7)^(e+1)-1)/(-p^7-1)]; f[2, e_] := 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 60] (* Amiram Eldar, Sep 27 2023 *)
-
PARI
apply( A321823(n)=sumdiv(n>>valuation(n,2),d,(2-d%4)*d^7), [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)^7*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^7)^(e+1)-1)/(p^7-1) if p == 1 (mod 4) and ((-p^7)^(e+1)-1)/(-p^7-1) if p == 3 (mod 4). - Amiram Eldar, Sep 27 2023
a(n) = Sum_{d|n} d^7*sin(d*Pi/2). - Ridouane Oudra, Aug 17 2024