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

1,1

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 <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...

Links

Table of n, a(n) for n=1..11.

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.

Example

2.0407097765...

Crossrefs

Cf. A005117, A013661, A076259.

Sequence in context: A046666 A276093 A176296 * A194795 A131575 A077957

Adjacent sequences: A349229 A349230 A349231 * A349233 A349234 A349235

Keyword

nonn,cons,more

Author

Amiram Eldar, Nov 11 2021

Status

approved