A366586 - OEIS

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

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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... .

LINKS

Table of n, a(n) for n=1..105.

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

Cf. A005117, A034444, A061742, A073184.

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

Adjacent sequences: A366583 A366584 A366585 * A366587 A366588 A366589

KEYWORD

nonn,cons

AUTHOR

Amiram Eldar, Oct 14 2023

STATUS

approved