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A053825
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Dirichlet inverse of sigma_3 function (A001158).
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11
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1, -9, -28, 8, -126, 252, -344, 0, 27, 1134, -1332, -224, -2198, 3096, 3528, 0, -4914, -243, -6860, -1008, 9632, 11988, -12168, 0, 125, 19782, 0, -2752, -24390, -31752, -29792, 0, 37296, 44226, 43344, 216, -50654, 61740, 61544, 0, -68922, -86688
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OFFSET
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1,2
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COMMENTS
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sigma_3(n) is the sum of the cubes of the divisors of n (A001158).
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REFERENCES
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Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 39.
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LINKS
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FORMULA
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Dirichlet g.f.: 1/(zeta(x)*zeta(x-3))
Multiplicative with a(p^1) = -1-p^3, a(p^2) = p^3, a(p^e) = 0 for e>=3. - Mitch Harris, Jun 27 2005
a(n) = Sum_{d divides n} d * A053822(d) * phi(n/d), where the totient function phi(n) = A000010(n).
a(n) = Sum_{d divides n} d^2 * (sigma_1(d))^(-1) * J_2(n/d) and
a(n) = Sum_{d divides n} d^3 * (sigma_k(d))^(-1) * J_(k+3)(n/d), where (sigma_k(n))^(-1) denotes the Dirichlet inverse of the divisor sum function sigma_k(n) and J_k(n) denotes the Jordan totient function. (End)
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MAPLE
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with(numtheory):seq(add( mobius(n/d)*mobius(d)*d^3, d in divisors(n)), n = 1..100); # Peter Bala, Jan 26 2024
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MATHEMATICA
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a[n_] := Sum[MoebiusMu[n/d] MoebiusMu[d] d^3, {d, Divisors[n]}];
f[p_, e_] := If[e == 1, -p^3 - 1, If[e == 2, p^3, 0]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 16 2020 *)
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PROG
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(PARI) seq(n)={dirdiv(vector(n, n, n==1), vector(n, n, sigma(n, 3)))} \\ Andrew Howroyd, Aug 05 2018
(PARI) for(n=1, 100, print1(direuler(p=2, n, (1 - X)*(1 - p^3*X))[n], ", ")) \\ Vaclav Kotesovec, Sep 16 2020
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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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