This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A349366 #13 Nov 20 2021 20:37:43
%S A349366 0,0,0,0,1,0,1,1,0,1,1,2,2,1,1,2,2,2,3,3,2,2,2,3,4,4,3,2,1,1,3,2,3,3,
%T A349366 4,3,3,4,3,4,5,4,4,3,2,1,4,5,4,4,3,3,3,4,5,5,5,4,3,2,2,4,4,4,3,2,4,4,
%U A349366 5,4,4,4,4,5,4,4,3,4,3,4,5,5,5,5,6,5
%N A349366 Number of primes p such that prime(n) < p <= prime(n) + (log(prime(n)))^2 - log(prime(n)) - 1.
%C A349366 This sequence is an example of the search for an elementary upper bound for prime gaps that is valid for all but finitely many cases. A182134 is motivated by Firoozbakht's conjecture. Kourbatov's paper proves that Firoozbakht's conjecture is equivalent to an upper bound on prime gaps of the form (log(p))^2 - log(p) - b, where 1 <= b <= 1.17. This sequence results from the choice b = 1. While Kourbatov's bound with b = 1 implies Firoozbakht's conjecture, the terms of this sequence appear to be smaller than A182134.
%C A349366 Conjectures: prime gaps are o((log(p))^2), but are larger infinitely often than (log(p))^(2 - epsilon), for any epsilon > 0.
%H A349366 Hal M. Switkay, <a href="/A349366/b349366.txt">Table of n, a(n) for n = 1..4210</a>
%H A349366 Alexei 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.
%e A349366 a(12) is the number of primes above prime(12), which is 37, in a gap whose width is (log(37))^2 + log(37) - 1 = 8.4278: that is, the number of primes between 37 and 45.4278, and that is 2 (namely, 41 and 43).
%t A349366 Table[Length@Select[Range[Prime@n+1,Prime@n+(Log[Prime@n])^2-Log[Prime@n]-1],PrimeQ],{n,86}] (* _Giorgos Kalogeropoulos_, Nov 15 2021 *)
%Y A349366 Cf. A182134.
%K A349366 nonn
%O A349366 1,12
%A A349366 _Hal M. Switkay_, Nov 15 2021