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
KEYWORD
nonn,mult
AUTHOR
N. J. A. Sloane, May 14 2011
STATUS
approved