A235492 Median of maximal "prime gaps" in Cramer's model with n urns.

1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11

Offset: 1

Author

Alexei Kourbatov, Jan 11 2014

Keywords

• nonn

Comments

In Cramer's probabilistic model of primes with n urns (Cramer, 1936, A235402), there exists a distribution of maximal "prime gaps". We can represent this distribution as a histogram. This sequence is the distribution's median, i.e. the (unique) x-coordinate of the histogram's bar with the following properties:

- the sum of this bar plus all bars to the left is 1/2 or more, AND

- the sum of this bar plus all bars to the right is 1/2 or more.

See A235402 for further comments.

Examples

			For n=3, the histogram bar at x=1 has the height 0.91>1/2. Therefore, x=1 is the histogram's median, so a(3)=1. See A235402 for more details.
		

Links

• H. Cramer, On the order of magnitude of the difference between consecutive prime numbers, Acta Arith. 2 (1936), 23-46.
• A. Kourbatov, The distribution of maximal prime gaps in Cramer's probabilistic model of primes, arXiv:1401.6959.
• A. Kourbatov, Maximal gaps between Cramer's random primes from 2 to N: cdf, histogram, mode, median
• A. Kourbatov, Upper bounds for prime gaps related to Firoozbakht's conjecture, J. Int. Seq. 18 (2015) 15.11.2

Crossrefs

Cf. A235402 (mode of maximal "prime gaps" in Cramer's model).

Formula

a(n) = n log(li n)/(li n) + O(n/li n), where li n is the logarithmic integral of n.