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

1,5

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.

Links

Table of n, a(n) for n=1..80.

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

Formula

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

Example

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.

Crossrefs

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

Sequence in context: A285881 A091373 A197637 * A226762 A347697 A300763

Adjacent sequences: A235489 A235490 A235491 * A235493 A235494 A235495

Keyword

nonn

Author

Alexei Kourbatov, Jan 11 2014

Status

approved