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A321832 a(n) = Sum_{d|n, n/d==1 (mod 4)} d^8 - Sum_{d|n, n/d==3 (mod 4)} d^8.

%I A321832 #26 Sep 27 2024 14:25:24
%S A321832 1,256,6560,65536,390626,1679360,5764800,16777216,43040161,100000256,
%T A321832 214358880,429916160,815730722,1475788800,2562506560,4294967296,
%U A321832 6975757442,11018281216,16983563040,25600065536,37817088000,54875873280,78310985280,110058536960,152588281251,208827064832
%N A321832 a(n) = Sum_{d|n, n/d==1 (mod 4)} d^8 - Sum_{d|n, n/d==3 (mod 4)} d^8.
%H A321832 Seiichi Manyama, <a href="/A321832/b321832.txt">Table of n, a(n) for n = 1..10000</a>
%H A321832 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 A321832 <a href="/index/Ge#Glaisher">Index entries for sequences mentioned by Glaisher</a>.
%F A321832 G.f.: Sum_{k>=1} k^8*x^k/(1 + x^(2*k)). - _Ilya Gutkovskiy_, Nov 26 2018
%F A321832 Multiplicative with a(p^e) = round(p^(8e+8)/(p^8 + (p mod 4) - 2)). (Following R. Israel in A321833.) - _M. F. Hasler_, Nov 26 2018
%F A321832 Sum_{k=1..n} a(k) ~ c * n^9 / 9, where c = 277*Pi^9/8257536 (A258816). - _Amiram Eldar_, Nov 04 2023
%F A321832 a(n) = Sum_{d|n} (n/d)^8*sin(d*Pi/2). - _Ridouane Oudra_, Sep 27 2024
%t A321832 s[n_,r_] := DivisorSum[n, #^8 &, Mod[n/#,4]==r &]; a[n_] := s[n,1] - s[n,3]; Array[a, 30] (* _Amiram Eldar_, Nov 26 2018 *)
%t A321832 s[n_] := If[OddQ[n], (-1)^((n-1)/2), 0]; (* A101455 *)
%t A321832 f[p_, e_] := (p^(8*e+8) - s[p]^(e+1))/(p^8 - s[p]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Nov 04 2023 *)
%o A321832 (PARI) apply( A321832(n)=factorback(apply(f->f[1]^(8*f[2]+8)\/(f[1]^8+f[1]%4-2),Col(factor(n)))), [1..50]) \\ _M. F. Hasler_, Nov 26 2018
%Y A321832 Cf. A101455, A258816.
%Y A321832 Cf. A321543 - A321565, A321807 - A321836 for similar sequences.
%Y A321832 Glaisher's E'_i (i=0..12): A002654, A050469, A050470, A050471, A050468, A321829, A321830, A321831, this sequence, A321833, A321834, A321835, A321836.
%K A321832 nonn,easy,mult
%O A321832 1,2
%A A321832 _N. J. A. Sloane_, Nov 24 2018