A190623 Mobius transform of A008457.

1, 6, 27, 64, 125, 162, 343, 512, 729, 750, 1331, 1728, 2197, 2058, 3375, 4096, 4913, 4374, 6859, 8000, 9261, 7986, 12167, 13824, 15625, 13182, 19683, 21952, 24389, 20250, 29791, 32768, 35937, 29478, 42875, 46656, 50653, 41154, 59319, 64000, 68921, 55566, 79507, 85184, 91125, 73002, 103823, 110592, 117649

Offset

1,2

Comments

Multiplicative because A008457 is. - Andrew Howroyd, Jul 23 2018

References

J. M. Borwein, D. H. Bailey and R. Girgensohn, Experimentation in Mathematics, A K Peters, Ltd., Natick, MA, 2004. x+357 pp. See p. 195.

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Formula

From Amiram Eldar, Dec 03 2022: (Start)

Multiplicative with a(2) = 6, a(2^e) = 8^e for e > 1, and a(p^e) = p^(3*e) for p > 2.

Dirichlet g.f.: zeta(s-3)*(1 - 2^(1-s) + 4^(2-s)).

Sum_{k=1..n} a(k) ~ (15/64) * n^4. (End)

Mathematica

b[n_] := (-1)^n Sum[(-1)^d d^3, {d, Divisors[n]}];
a[n_] := Sum[MoebiusMu[d] b[n/d], {d, Divisors[n]}];
Array[a, 49] (* Jean-François Alcover, Sep 07 2019, from PARI *)
f[p_, e_] := p^(3*e); f[2, 1] = 6; f[2, e_] := 8^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Dec 03 2022 *)

Prog

(PARI) \\ here b(n) is A008457.
b(n)=(-1)^n*sumdiv(n, d, (-1)^d*d^3);
a(n)=sumdiv(n, d, moebius(d)*b(n/d)); \\ Andrew Howroyd, Jul 23 2018

Crossrefs

Cf. A008457.

Sequence in context: A217189 A363696 A167469 * A305158 A273408 A085788

Adjacent sequences: A190620 A190621 A190622 * A190624 A190625 A190626

Keyword

nonn,mult

Author

N. J. A. Sloane, May 14 2011

Status

approved