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A321835 a(n) = Sum_{d|n, n/d==1 mod 4} d^11 - Sum_{d|n, n/d==3 mod 4} d^11.

%I A321835 #28 Sep 27 2024 16:21:17
%S A321835 1,2048,177146,4194304,48828126,362795008,1977326742,8589934592,
%T A321835 31380882463,100000002048,285311670610,743004176384,1792160394038,
%U A321835 4049565167616,8649707208396,17592186044416,34271896307634,64268047284224,116490258898218
%N A321835 a(n) = Sum_{d|n, n/d==1 mod 4} d^11 - Sum_{d|n, n/d==3 mod 4} d^11.
%H A321835 Seiichi Manyama, <a href="/A321835/b321835.txt">Table of n, a(n) for n = 1..10000</a>
%H A321835 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 A321835 <a href="/index/Ge#Glaisher">Index entries for sequences mentioned by Glaisher</a>.
%F A321835 G.f.: Sum_{k>=1} k^11*x^k/(1 + x^(2*k)). - _Ilya Gutkovskiy_, Nov 26 2018
%F A321835 From _Amiram Eldar_, Nov 04 2023: (Start)
%F A321835 Multiplicative with a(p^e) = (p^(11*e+11) - A101455(p)^(e+1))/(p^11 - A101455(p)).
%F A321835 Sum_{k=1..n} a(k) ~ c * n^12 / 12, where c = beta(12) = 0.99999812235..., and beta is the Dirichlet beta function. (End)
%F A321835 a(n) = Sum_{d|n} (n/d)^11*sin(d*Pi/2). - _Ridouane Oudra_, Sep 27 2024
%t A321835 s[n_,r_] := DivisorSum[n, #^11 &, Mod[n/#,4]==r &]; a[n_] := s[n,1] - s[n,3]; Array[a, 30] (* _Amiram Eldar_, Nov 26 2018 *)
%t A321835 s[n_] := If[OddQ[n], (-1)^((n-1)/2), 0]; (* A101455 *)
%t A321835 f[p_, e_] := (p^(11*e+11) - s[p]^(e+1))/(p^11 - s[p]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Nov 04 2023 *)
%o A321835 (PARI) apply( a(n)=sumdiv(n,d,if(bittest(n\d,0),(2-n\d%4)*d^11)), [1..30]) \\ _M. F. Hasler_, Nov 26 2018
%Y A321835 Cf. A101455.
%Y A321835 Cf. A321807 - A321836 for related sequences.
%Y A321835 Glaisher's E'_i (i=0..12): A002654, A050469, A050470, A050471, A050468, A321829, A321830, A321831, A321832, A321833, A321834, this sequence, A321836.
%K A321835 nonn,easy,mult
%O A321835 1,2
%A A321835 _N. J. A. Sloane_, Nov 24 2018