cp's OEIS Frontend

A246810 a(n) is the smallest number m such that np(m) = n, where np(m) is number of primes p such that prime(m) < p < prime(m)^(1 + 1/m).

%I A246810 #34 Jun 09 2018 10:43:58
%S A246810 1,5,12,17,25,55,83,169,207,206,384,953,1615,2192,2197,3024,3023,
%T A246810 10709,10935,29509,29508,62736,62735,94333,94332,196966,314940,608777,
%U A246810 1258688,1767259,2448975,2448973,7939362,9373136,9373134,16854966,16854967
%N A246810 a(n) is the smallest number m such that np(m) = n, where np(m) is number of primes p such that prime(m) < p < prime(m)^(1 + 1/m).
%C A246810 Firoozbakht's conjecture says that for every n, there exists at least one prime p where, prime(n) < p < prime(n)^(1 + 1/n). Hence if Firoozbakht's conjecture is true, then there is no m such that np(m) = 0.
%C A246810 Conjecture: For every positive integer n, a(n) exists.
%C A246810 a(65) > 10^12. - _Robert Price_, Nov 12 2014
%H A246810 Robert Price, <a href="/A246810/b246810.txt">Table of n, a(n) for n = 1..64</a>
%H A246810 A. Kourbatov, <a href="http://arxiv.org/abs/1503.01744">Verification of the Firoozbakht conjecture for primes up to four quintillion</a>, arXiv:1503.01744 [math.NT], 2015
%H A246810 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
%H A246810 Nilotpal Kanti Sinha, <a href="http://arxiv.org/abs/1010.1399">On a new property of primes that leads to a generalization of Cramer's conjecture</a>, arXiv:1010.1399v2 [math.NT], 2010.
%H A246810 Wikipedia, <a href="http://en.wikipedia.org/wiki/Firoozbakht%E2%80%99s_conjecture">Firoozbakht's conjecture</a>.
%e A246810 a(6) = 55 since the number of primes p such that prime(55) < p < prime(55)^(1 + 1/55) is 6 and 55 is the smallest number with this property.
%t A246810 np[n_]:=(b=Prime[n]; Length[Select[Range[b+1, b^(1 + 1/n)],PrimeQ]]); a[n_]:=(For[m=1, np[m] !=n, m++]; m);
%t A246810 Do[Print[a[n]], {n, 37}]
%Y A246810 Cf. A000040, A182134, A246781, A246782, A246783, A246787.
%K A246810 nonn
%O A246810 1,2
%A A246810 _Farideh Firoozbakht_ and _Jahangeer Kholdi_, Oct 10 2014