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 A321836 a(n) = Sum_{d|n, n/d==1 mod 4} d^12 - Sum_{d|n, n/d==3 mod 4} d^12. 58
 1, 4096, 531440, 16777216, 244140626, 2176778240, 13841287200, 68719476736, 282429005041, 1000000004096, 3138428376720, 8916083671040, 23298085122482, 56693912371200, 129746094281440, 281474976710656, 582622237229762, 1156829204647936 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 LINKS Seiichi Manyama, Table of n, a(n) for n = 1..10000 J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 4 and p. 8). Index entries for sequences mentioned by Glaisher. FORMULA G.f.: Sum_{k>=1} k^12*x^k/(1 + x^(2*k)). - Ilya Gutkovskiy, Nov 26 2018 From Amiram Eldar, Nov 04 2023: (Start) Multiplicative with a(p^e) = (p^(12*e+12) - A101455(p)^(e+1))/(p^12 - A101455(p)). 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) MATHEMATICA 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 *) s[n_] := If[OddQ[n], (-1)^((n-1)/2), 0]; (* A101455 *) 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 *) PROG (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 CROSSREFS Cf. A101455. Cf. A321543 - A321565, A321807 - A321835 for similar sequences. Glaisher's E'_i (i=0..12): A002654, A050469, A050470, A050471, A050468, A321829, A321830, A321831, A321832, A321833, A321834, A321835, this sequence. Sequence in context: A232961 A223966 A195660 * A016902 A017688 A008456 Adjacent sequences: A321833 A321834 A321835 * A321837 A321838 A321839 KEYWORD nonn,easy,mult AUTHOR N. J. A. Sloane, Nov 24 2018 STATUS approved

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Last modified November 30 03:08 EST 2023. Contains 367452 sequences. (Running on oeis4.)