A349232 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.

2, 0, 4, 0, 7, 0, 9, 7, 7, 6, 5

Offset: 1

Author

Amiram Eldar, Nov 11 2021

Keywords

• nonn
• cons
• more

Comments

Erdős (1951) proved the existence of this limit and Mossinghoff et al. (2021) calculated its first 11 decimal digits.

Let g(n) = A076259(n) be the sequence of gaps between squarefree numbers. The asymptotic mean of g is = Pi^2/6 (A013661). The second raw moment of g is = (P^2/6) * 2.0407097765... = 3.35683303..., the second central moment, or variance, of g is - ^2 = 0.651024947... and the standard deviation is sqrt( - ^2) = 0.8068611...

Examples

			2.0407097765...
		

Links

• Paul Erdős, Some problems and results in elementary number theory, Publ. Math. Debrecen, Vol. 2 (1951), pp. 103-109.
• 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.

Crossrefs

Cf. A005117, A013661, A076259.