A366587 Decimal expansion of the asymptotic mean of the ratio between the number of squarefree divisors and the number of cubefree divisors.

8, 5, 6, 2, 0, 0, 5, 0, 7, 9, 3, 7, 4, 7, 7, 1, 4, 9, 3, 9, 7, 2, 8, 1, 0, 8, 9, 5, 9, 5, 1, 6, 0, 4, 0, 4, 9, 8, 8, 4, 9, 0, 3, 1, 5, 8, 4, 1, 3, 2, 7, 1, 3, 1, 8, 5, 9, 6, 9, 5, 5, 8, 0, 3, 4, 0, 3, 8, 6, 6, 0, 8, 9, 6, 0, 1, 1, 9, 5, 9, 2, 1, 0, 5, 5, 5, 3, 0, 9, 0, 7, 8, 0, 9, 2, 3, 1, 4, 3, 4, 9, 2, 7, 3, 9

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... .

Links

Table of n, a(n) for n=0..104.

Formula

Equals lim_{m->oo} (1/m) * Sum_{k=1..m} A034444(k)/A073184(k).

Equals Product_{p prime} (1 - 1/(3*p^2)).

In general, the asymptotic mean of the ratio between the number of k-free divisors and the number of (k-1)-free divisors, for k >= 3, is Product_{p prime} (1 - 1/(k*p^2)).

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

Cf. A005117, A034444, A073184.

Similar constants: A307869, A308042, A308043, A358659, A361059, A361060, A361061, A361062, A366586 (mean of the inverse ratio).

Sequence in context: A318335 A243379 A214174 * A154433 A335339 A107828

Adjacent sequences: A366584 A366585 A366586 * A366588 A366589 A366590

Keyword

nonn,cons

Author

Amiram Eldar, Oct 14 2023

Status

approved