OFFSET
1,8
COMMENTS
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
FORMULA
a(n) = abs(A307428(n)).
Multiplicative with a(p) = 1 for e <= 2, and 2 for e >= 3.
a(n) >= 1, with equality if and only if n is cubefree (A004709).
Dirichlet g.f.: zeta(s)*zeta(3*s)/zeta(6*s).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = zeta(3)/zeta(6) = 1.181564... (A157289).
In general, the asymptotic mean of the number of unitary divisors of the k-full part of n is zeta(k)/zeta(2*k).
MATHEMATICA
f[p_, e_] := If[e > 2, 2, 1]; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
PROG
(PARI) a(n) = vecprod(apply(x -> if(x < 3, 1, 2), factor(n)[, 2]));
CROSSREFS
KEYWORD
nonn,easy,mult
AUTHOR
Amiram Eldar, Dec 19 2023
STATUS
approved