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A053826
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Dirichlet inverse of sigma_4 function (A001159).
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9
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1, -17, -82, 16, -626, 1394, -2402, 0, 81, 10642, -14642, -1312, -28562, 40834, 51332, 0, -83522, -1377, -130322, -10016, 196964, 248914, -279842, 0, 625, 485554, 0, -38432, -707282, -872644, -923522, 0, 1200644, 1419874, 1503652, 1296, -1874162
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OFFSET
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1,2
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COMMENTS
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sigma_4(n) is the sum of the 4th powers of the divisors of n (A001159).
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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-4)).
Multiplicative with a(p^1) = -1 - p^4, a(p^2) = p^4, a(p^e) = 0 for e>=3. - Mitch Harris, Jun 27 2005
a(n) = Sum_{d divides n} d * A053825(d) * phi(n/d), where the totient function phi(n) = A000010(n).
a(n) = Sum_{d divides n} d^2 * (sigma_2(d))^(-1) * J_2(n/d),
a(n) = Sum_{d divides n} d^3 * (sigma_1(d))^(-1) * J_3(n/d), and for k >= 0,
a(n) = Sum_{d divides n} d^4 * (sigma_k(d))^(-1) * J_(k+4)(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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MATHEMATICA
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Table[DivisorSum[n, MoebiusMu[n/#]*MoebiusMu[#]*#^4 &], {n, 1, 50}] (* G. C. Greubel, Nov 07 2018 *)
f[p_, e_] := If[e == 1, -p^4 - 1, If[e == 2, p^4, 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) a(n) = sumdiv(n, d, moebius(n/d)*moebius(d)*d^4); \\ Michel Marcus, Nov 06 2018
(PARI) for(n=1, 100, print1(direuler(p=2, n, (1 - X)*(1 - p^4*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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