A383652 -id:A383652 - cp's OEIS frontend

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

Giuseppe Coppoletta, Jun 19 2017

nonn

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

			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).

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

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

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

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

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

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

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

A381850 Primes p preceded and followed by primes whose difference is less than 2*log(p).

Original entry on oeis.org

41, 43, 59, 61, 71, 73, 101, 103, 107, 109, 137, 151, 163, 167, 179, 193, 197, 227, 229, 233, 239, 269, 271, 277, 281, 311, 313, 349, 353, 379, 383, 419, 421, 431, 433, 439, 443, 457, 461, 463, 487, 491, 499, 503, 563, 569, 571, 593, 599, 601, 607, 613, 617, 641, 643, 647, 653

Offset: 1

Alain Rocchelli, May 06 2025

nonn

Primes prime(k) such that prime(k+1) - prime(k-1) < 2*log(prime(k)).

Since the geometric mean is never greater than the arithmetic mean: this sequence is a subsequence of A383652.

			19 is not a term because 23-17=6 and 2*log(19)=5.8889.
41 is a term because 43-37=6 and 2*log(41)=7.4271.
131 is not a term because 137-127=10 and 2*log(131)=9.7504.
137 is a term because 139-131=8 and 2*log(137)=9.8400.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000040, A031131, A383652.

A288907 is a subsequence.

P:= select(isprime,[2,seq(i,i=3..1000,2)]):
P[select(i -> is(P[i+1]-P[i-1] < 2*log(P[i])), [$2..nops(P)-1])]; # Robert Israel, Jun 06 2025

Select[Prime[Range[120]],NextPrime[#] - NextPrime[#,-1] < 2Log[#] &] (* Stefano Spezia, May 06 2025 *)

forprime(P=3, 800, my(M=precprime(P-1), Q=nextprime(P+1)); if(Q-M<2*log(P), print1(P,", ")));

Conjecture: Limit_{n->oo} n / PrimePi(a(n)) = 1-(3/e^2).

cp's OEIS Frontend

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

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