

A369306


The number of cubefree divisors d of n such that n/d is also cubefree.


1



1, 2, 2, 3, 2, 4, 2, 2, 3, 4, 2, 6, 2, 4, 4, 1, 2, 6, 2, 6, 4, 4, 2, 4, 3, 4, 2, 6, 2, 8, 2, 0, 4, 4, 4, 9, 2, 4, 4, 4, 2, 8, 2, 6, 6, 4, 2, 2, 3, 6, 4, 6, 2, 4, 4, 4, 4, 4, 2, 12, 2, 4, 6, 0, 4, 8, 2, 6, 4, 8, 2, 6, 2, 4, 6, 6, 4, 8, 2, 2, 1, 4, 2, 12, 4, 4, 4
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OFFSET

1,2


COMMENTS

The analogous sequence with squarefree divisors (the number of squarefree divisors d of n such that n/d is also squarefree) is abs(A007427(n)).


LINKS



FORMULA

Multiplicative with a(p) = 2, a(p^2) = 3, a(p^3) = 2, a(p^4) = 1, and a(p^e) = 0 for e >= 5.
a(n) >= 0, with equality if and only if n is a 5full number (A069492) larger than 1.
a(n) = 1 if and only if n is the 4th power of a squarefree number (A005117).
a(n) <= A000005(n), with equality if and only if n is cubefree (A004709).
Dirichlet g.f.: zeta(s)^2/zeta(3*s)^2.
Sum_{k=1..n} a(k) ~ (n/zeta(3)^2) * (log(n) + 2*gamma  1  6*zeta'(3)/zeta(3)), where gamma is Euler's constant (A001620).


MATHEMATICA

f[p_, e_] := Switch[e, 1, 2, 2, 3, 3, 2, 4, 1, _, 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]


PROG

(PARI) a(n) = vecprod(apply(x > [2, 3, 2, 1, 0][min(x, 5)], factor(n)[, 2]));


CROSSREFS



KEYWORD

nonn,easy,mult


AUTHOR



STATUS

approved



