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

1,1

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

Links

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Eric Weisstein's World of Mathematics, Prime Number Theorem

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

Example

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).

Maple

q:= p-> isprime(p) and is(max(nextprime(p)-p, p-prevprime(p))<log(p)):
select(q, [$3..1200])[];  # Alois P. Heinz, May 12 2025

Mathematica

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

Prog

(SageMath) [n for n in prime_range(3, 1300) if next_prime(n)-n<log(n) and n-previous_prime(n)<log(n)] #
(PARI) is(n) = ispseudoprime(n) && n-precprime(n-1) < log(n) && nextprime(n+1)-n < log(n) \\ Felix Fröhlich, Jun 19 2017

Crossrefs

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

Sequence in context: A023282 A155953 A247200 * A234962 A166252 A339466

Adjacent sequences: A288904 A288905 A288906 * A288908 A288909 A288910

Keyword

nonn

Author

Giuseppe Coppoletta, Jun 19 2017

Status

approved