%I #8 Oct 14 2023 12:22:05

%S 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,

%T 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,

%U 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

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

%C 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).

%C r(k) >= 1 with equality if and only if k is squarefree (A005117).

%C 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.

%C 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... .

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

%F Equals Product_{p prime} (1 + 1/(2*p^2)).

%F 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)).

%e 1.24253418622467728695963000629433770800015253305890...

%t $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]]

%o (PARI) prodeulerrat(1 + 1/(2*p^2))

%Y Cf. A005117, A034444, A061742, A073184.

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

%K nonn,cons

%O 1,2

%A _Amiram Eldar_, Oct 14 2023