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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 = 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... 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

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