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

1,12

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.

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.

Example

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).

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 *)

Crossrefs

Cf. A182134.

Sequence in context: A053257 A331002 A358234 * A151702 A151552 A160418

Adjacent sequences: A349363 A349364 A349365 * A349367 A349368 A349369

Keyword

nonn

Author

Hal M. Switkay, Nov 15 2021

Status

approved