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

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

Author

Alain Rocchelli, May 06 2025

Keywords

• nonn

Comments

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.

Examples

			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.
		

Links

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

Crossrefs

Cf. A000040, A031131, A383652.

A288907 is a subsequence.

Programs

Maple

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

Mathematica

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

PARI

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

Formula

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