A349232 -id:A349232 - cp's OEIS frontend

A349237 Decimal expansion of lim_{x->oo} (1/x) * Sum_{c(k+1) <= x} (c(k+1) - c(k))^2, where c(k) = A004709(k) is the k-th cubefree number.

1, 3, 6, 3, 1, 2, 9, 8, 9, 8, 0

Offset: 1

Amiram Eldar, Nov 11 2021

Huxley (1997) proved the existence of this limit and Mossinghoff et al. (2021) calculated its first 11 decimal digits.

Let g(n) = A349236(n) be the sequence of gaps between cubefree numbers. The asymptotic mean of g is = zeta(3) (A002117). The second raw moment of g is = zeta(3) * 1.3631298980... = 1.638559703..., the second central moment, or variance, of g is - ^2 = 0.193618905... and the standard deviation is sqrt( - ^2) = 0.440021482...

			1.3631298980...

M. N. Huxley, Moments of differences between square-free numbers, in G. R. H. Greaves, G. Harman and M. N. Huxley (eds.), Sieve methods, exponential sums, and their applications in number theory (Cardiff, 1995), London Math. Soc. Lecture Note Series, Vol. 237, Cambridge Univ. Press, Cambridge, 1997, pp. 187-204.

Michael J. Mossinghoff, Tomás Oliveira e Silva, and Tim Trudgian, The distribution of k-free numbers, Mathematics of Computation, Vol. 90, No. 328 (2021), pp. 907-929; arXiv preprint, arXiv:1912.04972 [math.NT], 2019-2020.

Cf. A002117, A004709, A349232, A349236.

cp's OEIS Frontend

A349237 Decimal expansion of lim_{x->oo} (1/x) * Sum_{c(k+1) <= x} (c(k+1) - c(k))^2, where c(k) = A004709(k) is the k-th cubefree number.

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