

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 kth cubefree number.


0



1, 3, 6, 3, 1, 2, 9, 8, 9, 8, 0
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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 squarefree 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. 187204.


LINKS



EXAMPLE

1.3631298980...


CROSSREFS



KEYWORD



AUTHOR



STATUS

approved



