OFFSET
1,2
COMMENTS
For a positive integer k the ratio between the number of cubefree divisors and the number of squarefree divisors is r(k) = A073184(k)/A034444(k).
r(k) >= 1 with equality if and only if k is squarefree (A005117).
The indices of records of this ratio are the squares of primorial numbers (A061742), and the corresponding record values are r(A061742(k)) = (3/2)^k. Therefore, this ratio is unbounded.
The asymptotic second raw moment is <r(k)^2> = Product_{p prime} (1 + 5/(4*p^2)) = 1.67242666864454336962... and the asymptotic standard deviation is 0.35851843008068965078... .
FORMULA
EXAMPLE
1.24253418622467728695963000629433770800015253305890...
MATHEMATICA
$MaxExtraPrecision = 1000; m = 1000; c = LinearRecurrence[{0, -(1/2)}, {0, 1}, m]; RealDigits[Exp[NSum[Indexed[c, n] * PrimeZetaP[n]/n, {n, 2, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 105][[1]]
PROG
(PARI) prodeulerrat(1 + 1/(2*p^2))
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Amiram Eldar, Oct 14 2023
STATUS
approved