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A368248
The number of unitary divisors of the cubefull part of n (A360540).
2
1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1
OFFSET
1,8
COMMENTS
First differs from A061704 and A362852 at n = 64, and from A304327 at n = 72.
Also, the number of squarefree divisors of the cubefull part of n.
Also, the number of cubes of squarefree numbers (A062838) that divide n.
LINKS
FORMULA
a(n) = A034444(A360540(n)).
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).
a(n) <= A034444(n), with equality if and only if n is cubefull (A036966).
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]));
KEYWORD
nonn,easy,mult
AUTHOR
Amiram Eldar, Dec 19 2023
STATUS
approved