A366587 - OEIS

%I #6 Oct 14 2023 11:29:38

%S 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,

%T 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,

%U 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

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

%C 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).

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

%C 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... .

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

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

%F 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)).

%e 0.85620050793747714939728108959516040498849031584132...

%t $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]]

%o (PARI) prodeulerrat(1 - 1/(3*p^2))

%Y Cf. A005117, A034444, A073184.		

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

%K nonn,cons

%O 0,1

%A _Amiram Eldar_, Oct 14 2023