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A050471
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a(n) = Sum_{d|n, n/d=1 mod 4} d^3 - Sum_{d|n, n/d=3 mod 4} d^3.
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15
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1, 8, 26, 64, 126, 208, 342, 512, 703, 1008, 1330, 1664, 2198, 2736, 3276, 4096, 4914, 5624, 6858, 8064, 8892, 10640, 12166, 13312, 15751, 17584, 18980, 21888, 24390, 26208, 29790, 32768, 34580, 39312, 43092, 44992, 50654, 54864, 57148
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OFFSET
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1,2
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COMMENTS
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Multiplicative because it is the Dirichlet convolution of A000578 = n^3 and A101455 = [1 0 -1 0 1 0 -1 ...], which are both multiplicative. - Christian G. Bower, May 17 2005
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LINKS
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FORMULA
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Multiplicative with a(p^e) = (p^(3*e+3) - A101455(p)^(e+1))/(p^3 - A101455(p)).
Sum_{k=1..n} a(k) ~ c * n^4 / 4, where c = A175572. (End)
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MATHEMATICA
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s[n_] := If[OddQ[n], (-1)^((n-1)/2), 0]; (* A101455 *)
f[p_, e_] := (p^(3*e+3) - s[p]^(e+1))/(p^3 - 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) a(n) = sumdiv(n, d, d^3*(((n/d) % 4)==1)) - sumdiv(n, d, d^3*(((n/d) % 4)==3)); \\ Michel Marcus, Feb 16 2015
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CROSSREFS
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Glaisher's E'_i (i=0..12): A002654, A050469, A050470, this sequence, A050468, A321829, A321830, A321831, A321832, A321833, A321834, A321835, A321836.
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KEYWORD
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nonn,easy,mult
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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