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A368929
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Dirichlet g.f.: zeta(s-2)^2 * (1 - 2^(3-s)) / zeta(s).
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3
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1, -1, 17, -16, 49, -17, 97, -112, 225, -49, 241, -272, 337, -97, 833, -640, 577, -225, 721, -784, 1649, -241, 1057, -1904, 1825, -337, 2673, -1552, 1681, -833, 1921, -3328, 4097, -577, 4753, -3600, 2737, -721, 5729, -5488, 3361, -1649, 3697, -3856, 11025, -1057, 4417
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OFFSET
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1,3
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COMMENTS
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LINKS
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FORMULA
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Sum_{k=1..n} a(k) ~ log(2) * n^3 / (3*zeta(3)).
Multiplicative with a(2^e) = -(3*e-2)*2^(2*e-2), and a(p^e) = p^(2*e)*(1 + e*(1-1/p^2)) for an odd prime p. - Amiram Eldar, Jan 12 2024
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MATHEMATICA
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Table[Sum[Sum[d^2 * MoebiusMu[k/d], {d, Divisors[k]}] * (-1)^(n/k + 1) * n^2/k^2, {k, Divisors[n]}], {n, 1, 100}]
f[p_, e_] := p^(2*e)*(1 + e*(1 - 1/p^2)); f[2, e_] := -(3*e - 2)*2^(2*e - 2); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jan 12 2024 *)
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PROG
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(PARI) a(n) = {my(f = factor(n), p, e); prod(i = 1, #f~, p = f[i, 1]; e = f[i, 2]; if(p == 2, -(3*e-2)*2^(2*e-2), p^(2*e)*(1 + e*(1-1/p^2)))); } \\ Amiram Eldar, Jan 12 2024
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CROSSREFS
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KEYWORD
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sign,mult
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AUTHOR
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STATUS
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approved
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