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A321821 a(n) = Sum_{d|n, d==1 mod 4} d^5 - Sum_{d|n, d==3 mod 4} d^5.

%I A321821 #32 Jun 24 2024 08:48:30
%S A321821 1,1,-242,1,3126,-242,-16806,1,58807,3126,-161050,-242,371294,-16806,
%T A321821 -756492,1,1419858,58807,-2476098,3126,4067052,-161050,-6436342,-242,
%U A321821 9768751,371294,-14290100,-16806,20511150,-756492,-28629150,1,38974100
%N A321821 a(n) = Sum_{d|n, d==1 mod 4} d^5 - Sum_{d|n, d==3 mod 4} d^5.
%H A321821 Seiichi Manyama, <a href="/A321821/b321821.txt">Table of n, a(n) for n = 1..10000</a>
%H A321821 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 A321821 <a href="/index/Ge#Glaisher">Index entries for sequences mentioned by Glaisher</a>.
%F A321821 a(n) = a(A000265(n)). - _M. F. Hasler_, Nov 26 2018
%F A321821 G.f.: Sum_{k>=1} (-1)^(k-1)*(2*k - 1)^5*x^(2*k-1)/(1 - x^(2*k-1)). - _Ilya Gutkovskiy_, Dec 22 2018
%F A321821 Multiplicative with a(2^e) = 1, and for an odd prime p, ((p^5)^(e+1)-1)/(p^5-1) if p == 1 (mod 4) and ((-p^5)^(e+1)-1)/(-p^5-1) if p == 3 (mod 4). - _Amiram Eldar_, Sep 27 2023
%F A321821 a(n) = Sum_{d|n} d^5*sin(d*Pi/2). - _Ridouane Oudra_, Jun 23 2024
%t A321821 s[n_, r_] := DivisorSum[n, #^5 &, Mod[#, 4] == r &]; a[n_] := s[n, 1] - s[n, 3]; Array[a, 30] (* _Amiram Eldar_, Nov 26 2018 *)
%t A321821 Table[With[{d=Divisors[n]},Total[Select[d,Mod[#,4]==1&]^5]-Total[Select[ d,Mod[ #,4]==3&]^5]],{n,40}] (* _Harvey P. Dale_, Dec 15 2018 *)
%t A321821 f[p_, e_] := If[Mod[p, 4] == 1, ((p^5)^(e+1)-1)/(p^5-1), ((-p^5)^(e+1)-1)/(-p^5-1)]; f[2, e_] := 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 60] (* _Amiram Eldar_, Sep 27 2023 *)
%o A321821 (PARI) apply( A321821(n)=sumdiv(n>>valuation(n,2),d,(2-d%4)*d^5), [1..40]) \\ _M. F. Hasler_, Nov 26 2018
%Y A321821 Column k=5 of A322143.
%Y A321821 Cf. A321543 - A321565, A321807 - A321836 for similar sequences.
%Y A321821 Cf. A000265.
%K A321821 sign,easy,mult
%O A321821 1,3
%A A321821 _N. J. A. Sloane_, Nov 24 2018