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A321825 a(n) = Sum_{d|n, d==1 (mod 4)} d^9 - Sum_{d|n, d==3 (mod 4)} d^9.

%I A321825 #32 Aug 19 2024 12:16:19
%S A321825 1,1,-19682,1,1953126,-19682,-40353606,1,387400807,1953126,
%T A321825 -2357947690,-19682,10604499374,-40353606,-38441425932,1,118587876498,
%U A321825 387400807,-322687697778,1953126,794239673292,-2357947690,-1801152661462,-19682,3814699218751
%N A321825 a(n) = Sum_{d|n, d==1 (mod 4)} d^9 - Sum_{d|n, d==3 (mod 4)} d^9.
%H A321825 Seiichi Manyama, <a href="/A321825/b321825.txt">Table of n, a(n) for n = 1..10000</a>
%H A321825 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 A321825 <a href="/index/Ge#Glaisher">Index entries for sequences mentioned by Glaisher</a>.
%F A321825 a(n) = a(A000265(n)). - _M. F. Hasler_, Nov 26 2018
%F A321825 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
%F A321825 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
%F A321825 a(n) = Sum_{d|n} d^9*sin(d*Pi/2). - _Ridouane Oudra_, Aug 18 2024
%t A321825 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 *)
%t A321825 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 *)
%o A321825 (PARI) apply( A321825(n)=sumdiv(n>>valuation(n,2),d,(2-d%4)*d^9), [1..40]) \\ _M. F. Hasler_, Nov 26 2018
%Y A321825 Column k=9 of A322143.
%Y A321825 Cf. A321543 - A321565, A321807 - A321836 for similar sequences.
%Y A321825 Cf. A000265.
%K A321825 sign,easy,mult
%O A321825 1,3
%A A321825 _N. J. A. Sloane_, Nov 24 2018