A288907 Primes p whose distance from the next prime and from the previous prime is less than log(p).

71, 101, 103, 107, 109, 193, 197, 227, 229, 281, 311, 313, 349, 433, 439, 443, 461, 463, 503, 563, 569, 571, 593, 599, 601, 607, 613, 617, 643, 647, 653, 659, 677, 733, 739, 757, 823, 827, 857, 859, 881, 883, 941, 947, 971, 977, 1013, 1019, 1033, 1063, 1091, 1093

Offset: 1

Author

Giuseppe Coppoletta, Jun 19 2017

Keywords

• nonn

Comments

Primes preceded and followed by less-than-average prime gaps (by the Prime Number Theorem, see link).

This sequence is a subsequence of A381850 and of A383652. - Alain Rocchelli, May 07 2025

Examples

			n = 23 is not a term because 23 - 19 > log(23) = 3.13...
n = 71 is a term because log(71) = 4.71.. and 73 - log(71) < 71 < 67 + log(71).
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Prime Number Theorem

Crossrefs

Cf. A151799, A151800, A288908, A068996, A381850, A383652.

Programs

Maple

q:= p-> isprime(p) and is(max(nextprime(p)-p, p-prevprime(p))Alois P. Heinz, May 12 2025

Mathematica

Select[Range[2, 220] // Prime, Max[ Abs[# - NextPrime[#, {-1, 1}]]] < Log[#] &] (* Giovanni Resta, Jun 19 2017 *)

PARI

is(n) = ispseudoprime(n) && n-precprime(n-1) < log(n) && nextprime(n+1)-n < log(n) \\ Felix Fröhlich, Jun 19 2017

Sage

[n for n in prime_range(3,1300) if next_prime(n)-n
				

Formula

A151800(a(n)) - log(a(n)) < a(n) < A151799(a(n)) + log(a(n)).

Conjecture: Limit_{n->oo} n / PrimePi(a(n)) = (1-1/e)^2 (A068996). - Alain Rocchelli, May 07 2025