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

1,2

Comments

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 <g> = zeta(3) (A002117). The second raw moment of g is <g^2> = zeta(3) * 1.3631298980... = 1.638559703..., the second central moment, or variance, of g is <g^2> - <g>^2 = 0.193618905... and the standard deviation is sqrt(<g^2> - <g>^2) = 0.440021482...

References

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.

Links

Table of n, a(n) for n=1..11.

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.

Example

1.3631298980...

Crossrefs

Cf. A002117, A004709, A349232, A349236.

Sequence in context: A200478 A340263 A256158 * A123060 A204931 A178817

Adjacent sequences: A349234 A349235 A349236 * A349238 A349239 A349240

Keyword

nonn,cons,more

Author

Amiram Eldar, Nov 11 2021

Status

approved