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A321836
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a(n) = Sum_{d|n, n/d==1 mod 4} d^12 - Sum_{d|n, n/d==3 mod 4} d^12.
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58
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1, 4096, 531440, 16777216, 244140626, 2176778240, 13841287200, 68719476736, 282429005041, 1000000004096, 3138428376720, 8916083671040, 23298085122482, 56693912371200, 129746094281440, 281474976710656, 582622237229762, 1156829204647936
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OFFSET
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1,2
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LINKS
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FORMULA
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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)
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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Glaisher's E'_i (i=0..12): A002654, A050469, A050470, A050471, A050468, A321829, A321830, A321831, A321832, A321833, A321834, A321835, this sequence.
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KEYWORD
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nonn,easy,mult
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AUTHOR
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STATUS
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approved
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