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A321833 a(n) = Sum_{d|n, n/d==1 mod 4} d^9 - Sum_{d|n, n/d==3 mod 4} d^9.

%I A321833 #29 Sep 27 2024 14:25:56
%S A321833 1,512,19682,262144,1953126,10077184,40353606,134217728,387400807,
%T A321833 1000000512,2357947690,5159518208,10604499374,20661046272,38441425932,
%U A321833 68719476736,118587876498,198349213184,322687697778,512000262144,794239673292
%N A321833 a(n) = Sum_{d|n, n/d==1 mod 4} d^9 - Sum_{d|n, n/d==3 mod 4} d^9.
%H A321833 Seiichi Manyama, <a href="/A321833/b321833.txt">Table of n, a(n) for n = 1..10000</a>
%H A321833 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 A321833 <a href="/index/Ge#Glaisher">Index entries for sequences mentioned by Glaisher</a>.
%F A321833 G.f.: Sum_{k>=1} k^9*x^k/(1 + x^(2*k)). - _Ilya Gutkovskiy_, Nov 26 2018
%F A321833 From _Robert Israel_, Nov 26 2018: (Start) a(2^m) = 2^(9*m).
%F A321833 For prime p == 1 (mod 4), a(p^m) = (p^(9(m+1))-1)/(p^9-1).
%F A321833 For prime p == 3 (mod 4), a(p^m) = (p^(9(m+1))+(-1)^m)/(p^9+1). (End)
%F A321833 From _Amiram Eldar_, Nov 04 2023: (Start)
%F A321833 Multiplicative with a(p^e) = (p^(9*e+9) - A101455(p)^(e+1))/(p^9 - A101455(p)).
%F A321833 Sum_{k=1..n} a(k) ~ c * n^10 / 10, where c = beta(10) = 0.99998316402... and beta is the Dirichlet beta function. (End)
%F A321833 a(n) = Sum_{d|n} (n/d)^9*sin(d*Pi/2). - _Ridouane Oudra_, Sep 27 2024
%p A321833 f:= n ->
%p A321833 mul(piecewise(t[1]=2,2^(9*t[2]), t[1] mod 4 = 1, (t[1]^(9*(t[2]+1))-1)/(t[1]^9-1), (t[1]^(9*(t[2]+1))+(-1)^t[2])/(t[1]^9+1)), t = ifactors(n)[2]):
%p A321833 map(f, [$1..100]); # _Robert Israel_, Nov 26 2018
%t A321833 s[n_,r_] := DivisorSum[n, #^9 &, Mod[n/#,4]==r &]; a[n_] := s[n,1] - s[n,3]; Array[a, 30] (* _Amiram Eldar_, Nov 26 2018 *)
%t A321833 s[n_] := If[OddQ[n], (-1)^((n-1)/2), 0]; (* A101455 *)
%t A321833 f[p_, e_] := (p^(9*e+9) - s[p]^(e+1))/(p^9 - s[p]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Nov 04 2023 *)
%o A321833 (PARI) apply( a(n)=sumdiv(n,d,if(bittest(n\d,0),(2-n\d%4)*d^9)), [1..30]) \\ _M. F. Hasler_, Nov 26 2018
%Y A321833 Cf. A101455.
%Y A321833 Cf. A321807 - A321836 for similar sequences.
%Y A321833 Glaisher's E'_i (i=0..12): A002654, A050469, A050470, A050471, A050468, A321829, A321830, A321831, A321832, this sequence, A321834, A321835, A321836.
%K A321833 nonn,easy,mult
%O A321833 1,2
%A A321833 _N. J. A. Sloane_, Nov 24 2018