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A369101
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Dirichlet g.f.: zeta(s-3)^2 * (1 - 2^(4-s)) / zeta(s).
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0
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1, -1, 53, -64, 249, -53, 685, -960, 2133, -249, 2661, -3392, 4393, -685, 13197, -11264, 9825, -2133, 13717, -15936, 36305, -2661, 24333, -50880, 46625, -4393, 76545, -43840, 48777, -13197, 59581, -118784, 141033, -9825, 170565, -136512, 101305, -13717, 232829
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OFFSET
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1,3
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COMMENTS
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In general, for k > 0, if Dirichlet g.f. is zeta(s-k)^2 * (1 - 2^(k+1-s)) / zeta(s), then a(n) ~ log(2) * n^(k+1) / ((k+1) * zeta(k+1)).
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LINKS
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FORMULA
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Sum_{k=1..n} a(k) ~ 45 * log(2) * n^4 / (2*Pi^4).
Multiplicative with a(2^e) = -(7*e-6)*8^(e-1), and a(p^e) = p^(3*e-3) * (1 + (e+1)*(p^3-1)) for an odd prime p. - Amiram Eldar, Jan 13 2024
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MATHEMATICA
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Table[Sum[DivisorSum[k, #^3*MoebiusMu[k/#]&]*(-1)^(n/k+1)*(n/k)^3, {k, Divisors[n]}], {n, 1, 50}]
f[p_, e_] := p^(3*e-3) * (1 + (e+1)*(p^3-1)); f[2, e_] := -(7*e-6)*8^(e-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jan 13 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, -(7*e-6)*8^(e-1), p^(3*e-3) * (1 + (e+1)*(p^3-1)))); } \\ Amiram Eldar, Jan 13 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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