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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. 0
1, 3, 6, 3, 1, 2, 9, 8, 9, 8, 0 (list; constant; graph; refs; listen; history; text; internal format)
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
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
Sequence in context: A200478 A340263 A256158 * A123060 A204931 A178817
KEYWORD
nonn,cons,more
AUTHOR
Amiram Eldar, Nov 11 2021
STATUS
approved

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Last modified April 21 01:42 EDT 2024. Contains 371850 sequences. (Running on oeis4.)