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%I #26 Nov 05 2023 09:03:52
%S 1,4096,531440,16777216,244140626,2176778240,13841287200,68719476736,
%T 282429005041,1000000004096,3138428376720,8916083671040,
%U 23298085122482,56693912371200,129746094281440,281474976710656,582622237229762,1156829204647936
%N a(n) = Sum_{d|n, n/d==1 mod 4} d^12 - Sum_{d|n, n/d==3 mod 4} d^12.
%H Seiichi Manyama, <a href="/A321836/b321836.txt">Table of n, a(n) for n = 1..10000</a>
%H J. W. L. Glaisher, <a href="https://books.google.com/books?id=bLs9AQAAMAAJ&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 <a href="/index/Ge#Glaisher">Index entries for sequences mentioned by Glaisher</a>.
%F G.f.: Sum_{k>=1} k^12*x^k/(1 + x^(2*k)). - _Ilya Gutkovskiy_, Nov 26 2018
%F From _Amiram Eldar_, Nov 04 2023: (Start)
%F Multiplicative with a(p^e) = (p^(12*e+12) - A101455(p)^(e+1))/(p^12 - A101455(p)).
%F Sum_{k=1..n} a(k) ~ c * n^13 / 13, where c = beta(13) = 540553*Pi^13/1569592442880 = 0.999999373583..., and beta is the Dirichlet beta function. (End)
%t s[n_,r_] := DivisorSum[n, #^12 &, Mod[n/#,4]==r &]; a[n_] := s[n,1] - s[n,3]; Array[a, 30] (* _Amiram Eldar_, Nov 26 2018 *)
%t s[n_] := If[OddQ[n], (-1)^((n-1)/2), 0]; (* A101455 *)
%t f[p_, e_] := (p^(12*e+12) - s[p]^(e+1))/(p^12 - s[p]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Nov 04 2023 *)
%o (PARI) apply( a(n)=sumdiv(n, d, if(bittest(n\d,0),(2-n\d%4)*d^12)), [1..30]) \\ _M. F. Hasler_, Nov 26 2018
%Y Cf. A101455.
%Y Cf. A321543 - A321565, A321807 - A321835 for similar sequences.
%Y Glaisher's E'_i (i=0..12): A002654, A050469, A050470, A050471, A050468, A321829, A321830, A321831, A321832, A321833, A321834, A321835, this sequence.
%K nonn,easy,mult
%O 1,2
%A _N. J. A. Sloane_, Nov 24 2018
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