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

Author

N. J. A. Sloane, May 14 2011

Keywords

• nonn
• mult

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

Crossrefs

Cf. A008457.

Programs

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 *)

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

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)