A349366 Number of primes p such that prime(n) < p <= prime(n) + (log(prime(n)))^2 - log(prime(n)) - 1.

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, 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, 5, 4, 4, 4, 4, 5, 4, 4, 3, 4, 3, 4, 5, 5, 5, 5, 6, 5

Offset: 1

Author

Hal M. Switkay, Nov 15 2021

Keywords

• nonn

Comments

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.

Conjectures: prime gaps are o((log(p))^2), but are larger infinitely often than (log(p))^(2 - epsilon), for any epsilon > 0.

Examples

			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).
		

Links

• Hal M. Switkay, Table of n, a(n) for n = 1..4210
• Alexei Kourbatov, Upper bounds for prime gaps related to Firoozbakht's conjecture, J. Int. Seq. 18 (2015) 15.11.2.

Crossrefs

Cf. A182134.

Programs

Mathematica

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 *)