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

%I A235492 #24 Jul 23 2025 08:34:43
%S A235492 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,
%T A235492 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,
%U A235492 10,10,10,10,10,11,11,11,11,11,11,11,11,11,11
%N A235492 Median of maximal "prime gaps" in Cramer's model with n urns.
%C A235492 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:
%C A235492 - the sum of this bar plus all bars to the left is 1/2 or more, AND
%C A235492 - the sum of this bar plus all bars to the right is 1/2 or more.
%C A235492 See A235402 for further comments.
%H A235492 H. Cramer, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa2/aa212.pdf">On the order of magnitude of the difference between consecutive prime numbers</a>, Acta Arith. 2 (1936), 23-46.
%H A235492 A. Kourbatov, <a href="http://arxiv.org/abs/1401.6959">The distribution of maximal prime gaps in Cramer's probabilistic model of primes</a>, arXiv:1401.6959.
%H A235492 A. Kourbatov, <a href="http://www.javascripter.net/math/statistics/maximalprimegapsincramermodel.htm">Maximal gaps between Cramer's random primes from 2 to N: cdf, histogram, mode, median</a>
%H A235492 A. Kourbatov, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Kourbatov/kourb7.html">Upper bounds for prime gaps related to Firoozbakht's conjecture</a>, J. Int. Seq. 18 (2015) 15.11.2
%F A235492 a(n) = n log(li n)/(li n) + O(n/li n), where li n is the logarithmic integral of n.
%e A235492 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.
%Y A235492 Cf. A235402 (mode of maximal "prime gaps" in Cramer's model).
%K A235492 nonn
%O A235492 1,5
%A A235492 _Alexei Kourbatov_, Jan 11 2014