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A366586
Decimal expansion of the asymptotic mean of the ratio between the number of cubefree divisors and the number of squarefree divisors.
2
1, 2, 4, 2, 5, 3, 4, 1, 8, 6, 2, 2, 4, 6, 7, 7, 2, 8, 6, 9, 5, 9, 6, 3, 0, 0, 0, 6, 2, 9, 4, 3, 3, 7, 7, 0, 8, 0, 0, 0, 1, 5, 2, 5, 3, 3, 0, 5, 8, 9, 0, 5, 9, 8, 0, 1, 9, 8, 3, 2, 2, 6, 8, 4, 7, 1, 5, 9, 2, 4, 7, 4, 4, 9, 2, 0, 0, 5, 9, 2, 9, 5, 1, 5, 5, 5, 2, 8, 3, 3, 0, 5, 8, 6, 2, 6, 6, 4, 9, 1, 9, 2, 9, 0, 6
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
Equals lim_{m->oo} (1/m) * Sum_{k=1..m} A073184(k)/A034444(k).
Equals Product_{p prime} (1 + 1/(2*p^2)).
In general, the asymptotic mean of the ratio between the number of (k+1)-free divisors and the number of k-free divisors, for k >= 2, is Product_{p prime} (1 + 1/(k*p^2)).
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
Similar constants: A307869, A308042, A308043, A358659, A361059, A361060, A361061, A361062, A366587 (mean of the inverse ratio).
Sequence in context: A057037 A076920 A183225 * A020774 A291303 A347098
KEYWORD
nonn,cons
AUTHOR
Amiram Eldar, Oct 14 2023
STATUS
approved