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A321826 a(n) = Sum_{d|n, d==1 mod 4} d^10 - Sum_{d|n, d==3 mod 4} d^10.

%I A321826 #28 Sep 25 2024 11:24:04
%S A321826 1,1,-59048,1,9765626,-59048,-282475248,1,3486725353,9765626,
%T A321826 -25937424600,-59048,137858491850,-282475248,-576640684048,1,
%U A321826 2015993900450,3486725353,-6131066257800,9765626,16679598443904,-25937424600,-41426511213648,-59048,95367441406251,137858491850
%N A321826 a(n) = Sum_{d|n, d==1 mod 4} d^10 - Sum_{d|n, d==3 mod 4} d^10.
%H A321826 Seiichi Manyama, <a href="/A321826/b321826.txt">Table of n, a(n) for n = 1..10000</a>
%H A321826 J. W. L. Glaisher, <a href="https://books.google.com/books?id=bLs9AQAAMAAJ&amp;pg=RA1-PA1">On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares</a>, Quart. J. Math. 38 (1907), 1-62 (see p. 4 and p. 8).
%H A321826 <a href="/index/Ge#Glaisher">Index entries for sequences mentioned by Glaisher</a>.
%F A321826 a(n) = a(A000265(n)). - _M. F. Hasler_, Nov 26 2018
%F A321826 G.f.: Sum_{k>=1} (-1)^(k-1)*(2*k - 1)^10*x^(2*k-1)/(1 - x^(2*k-1)). - _Ilya Gutkovskiy_, Dec 22 2018
%F A321826 Multiplicative with a(2^e) = 1, and for an odd prime p, ((p^10)^(e+1)-1)/(p^10-1) if p == 1 (mod 4) and ((-p^10)^(e+1)-1)/(-p^10-1) if p == 3 (mod 4). - _Amiram Eldar_, Sep 27 2023
%F A321826 a(n) = Sum_{d|n} d^10*sin(d*Pi/2). - _Ridouane Oudra_, Sep 04 2024
%t A321826 s[n_, r_] := DivisorSum[n, #^10 &, Mod[#, 4] == r &]; a[n_] := s[n, 1] - s[n, 3]; Array[a, 30] (* _Amiram Eldar_, Nov 26 2018 *)
%t A321826 f[p_, e_] := If[Mod[p, 4] == 1, ((p^10)^(e+1)-1)/(p^10-1), ((-p^10)^(e+1)-1)/(-p^10-1)]; f[2, e_] := 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 60] (* _Amiram Eldar_, Sep 27 2023 *)
%o A321826 (PARI) apply( A321826(n)=sumdiv(n>>valuation(n,2),d,(2-d%4)*d^10), [1..40]) \\ _M. F. Hasler_, Nov 26 2018
%Y A321826 Column k=10 of A322143.
%Y A321826 Cf. A321543 - A321565, A321807 - A321836 for similar sequences.
%Y A321826 Cf. A000265.
%K A321826 sign,easy,mult
%O A321826 1,3
%A A321826 _N. J. A. Sloane_, Nov 24 2018