OFFSET
0,1
COMMENTS
For a positive integer k the ratio between the number of squarefree divisors and the number of cubefree divisors is r(k) = A034444(k)/A073184(k).
r(k) <= 1 with equality if and only if k is squarefree (A005117).
The asymptotic second raw moment is <r(k)^2> = Product_{p prime} (1 - 5/(9*p^2)) = 0.76780883634140395932... and the asymptotic standard deviation is 0.29730736888962774256... .
FORMULA
EXAMPLE
0.85620050793747714939728108959516040498849031584132...
MATHEMATICA
$MaxExtraPrecision = 1000; m = 1000; c = LinearRecurrence[{0, 1/3}, {0, -(2/3)}, 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/(3*p^2))
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Amiram Eldar, Oct 14 2023
STATUS
approved