A366586 Decimal expansion of the asymptotic mean of the ratio between the number of cubefree divisors and the number of squarefree divisors.

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

Author

Amiram Eldar, Oct 14 2023

Keywords

• nonn
• cons

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 = Product_{p prime} (1 + 5/(4*p^2)) = 1.67242666864454336962... and the asymptotic standard deviation is 0.35851843008068965078... .

Examples

			1.24253418622467728695963000629433770800015253305890...
		

Crossrefs

Cf. A005117, A034444, A061742, A073184.

Similar constants: A307869, A308042, A308043, A358659, A361059, A361060, A361061, A361062, A366587 (mean of the inverse ratio).

Programs

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]]

PARI

prodeulerrat(1 + 1/(2*p^2))

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)).