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A321834 a(n) = Sum_{d|n, n/d==1 mod 4} d^10 - Sum_{d|n, n/d==3 mod 4} d^10.

%I A321834 #22 Sep 27 2024 16:20:41
%S A321834 1,1024,59048,1048576,9765626,60465152,282475248,1073741824,
%T A321834 3486725353,10000001024,25937424600,61916315648,137858491850,
%U A321834 289254653952,576640684048,1099511627776,2015993900450,3570406761472,6131066257800,10240001048576
%N A321834 a(n) = Sum_{d|n, n/d==1 mod 4} d^10 - Sum_{d|n, n/d==3 mod 4} d^10.
%H A321834 Seiichi Manyama, <a href="/A321834/b321834.txt">Table of n, a(n) for n = 1..10000</a>
%H A321834 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 A321834 <a href="/index/Ge#Glaisher">Index entries for sequences mentioned by Glaisher</a>.
%F A321834 G.f.: Sum_{k>=1} k^10*x^k/(1 + x^(2*k)). - _Ilya Gutkovskiy_, Nov 26 2018
%F A321834 From _Amiram Eldar_, Nov 04 2023: (Start)
%F A321834 Multiplicative with a(p^e) = (p^(10*e+10) - A101455(p)^(e+1))/(p^10 - A101455(p)).
%F A321834 Sum_{k=1..n} a(k) ~ c * n^11 / 11, where c = beta(11) = 50521*Pi^11/14863564800 = 0.999994374973... and beta is the Dirichlet beta function. (End)
%F A321834 a(n) = Sum_{d|n} (n/d)^10*sin(d*Pi/2). - _Ridouane Oudra_, Sep 27 2024
%t A321834 s[n_,r_] := DivisorSum[n, #^10 &, Mod[n/#,4]==r &]; a[n_] := s[n,1] - s[n,3]; Array[a, 30] (* _Amiram Eldar_, Nov 26 2018 *)
%t A321834 s[n_] := If[OddQ[n], (-1)^((n-1)/2), 0]; (* A101455 *)
%t A321834 f[p_, e_] := (p^(10*e+10) - s[p]^(e+1))/(p^10 - s[p]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Nov 04 2023 *)
%o A321834 (PARI) apply( a(n)=sumdiv(n, d, if(bittest(n\d,0),(2-n\d%4)*d^10)), [1..30]) \\ _M. F. Hasler_, Nov 26 2018
%Y A321834 Cf. A101455.
%Y A321834 Cf. A321807 - A321836 for similar sequences.
%Y A321834 Glaisher's E'_i (i=0..12): A002654, A050469, A050470, A050471, A050468, A321829, A321830, A321831, A321832, A321833, this sequence, A321835, A321836.
%K A321834 nonn,easy,mult
%O A321834 1,2
%A A321834 _N. J. A. Sloane_, Nov 24 2018