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%I #5 Nov 11 2021 20:06:48
%S 2,0,4,0,7,0,9,7,7,6,5
%N Decimal expansion of lim_{x->oo} (1/x) * Sum_{s(k+1) <= x} (s(k+1) - s(k))^2, where s(k) = A005117(k) is the k-th squarefree number.
%C Erdős (1951) proved the existence of this limit and Mossinghoff et al. (2021) calculated its first 11 decimal digits.
%C Let g(n) = A076259(n) be the sequence of gaps between squarefree numbers. The asymptotic mean of g is <g> = Pi^2/6 (A013661). The second raw moment of g is <g^2> = (P^2/6) * 2.0407097765... = 3.35683303..., the second central moment, or variance, of g is <g^2> - <g>^2 = 0.651024947... and the standard deviation is sqrt(<g^2> - <g>^2) = 0.8068611...
%H Paul Erdős, <a href="https://old.renyi.hu/~p_erdos/1951-13.pdf">Some problems and results in elementary number theory</a>, Publ. Math. Debrecen, Vol. 2 (1951), pp. 103-109.
%H Michael J. Mossinghoff, Tomás Oliveira e Silva, and Tim Trudgian, <a href="https://doi.org/10.1090/mcom/3581">The distribution of k-free numbers</a>, Mathematics of Computation, Vol. 90, No. 328 (2021), pp. 907-929; <a href="https://arxiv.org/abs/1912.04972">arXiv preprint</a>, arXiv:1912.04972 [math.NT], 2019-2020.
%e 2.0407097765...
%Y Cf. A005117, A013661, A076259.
%K nonn,cons,more
%O 1,1
%A _Amiram Eldar_, Nov 11 2021