A384618 Primes preceded and followed by gaps whose quotient (value greater or equal to 1) is greater than 2.

29, 31, 59, 61, 73, 113, 127, 137, 139, 149, 151, 179, 181, 191, 199, 223, 239, 241, 269, 271, 283, 307, 317, 331, 347, 419, 421, 431, 433, 467, 521, 523, 541, 569, 571, 599, 601, 619, 641, 659, 661, 673, 773, 809, 811, 821, 829, 853, 863, 877, 887, 907, 953, 967

Offset: 1

Author

Alain Rocchelli, Jun 04 2025

Keywords

• nonn

Comments

Primes prime(k) such that Max(prime(k)-prime(k-1),prime(k+1)-prime(k)) / Min(prime(k)-prime(k-1),prime(k+1)-prime(k)) > 2.

Examples

			19 is not a term because Max(19-17,23-19)/Min(19-17,23-19) = 4/2 = 2.
29 is a term because Max(29-23,31-29)/Min(29-23,31-29) = 6/2 = 3.
31 is a term because Max(31-29,37-31)/Min(31-29,37-31) = 6/2 = 3.
37 is not a term because Max(37-31,41-37)/Min(37-31,41-37) = 6/4 = 1.5.
		

Crossrefs

Cf. A384603.

Programs

PARI

forprime(P=3, 1000, my(M=P-precprime(P-1), Q=nextprime(P+1)-P, AR=max(M, Q)/min(M, Q), AR0=2); if(AR>AR0, print1(P, ", ")));

Python

from itertools import islice
from sympy import nextprime
def A384618_gen(): # generator of terms
    p,q,r = 2,3,5
    while True:
        s, t = q-p, r-q
        if s>(t<<1) or t>(s<<1): yield q
        p, q, r = q, r, nextprime(r)
A384618_list = list(islice(A384618_gen(),54)) # Chai Wah Wu, Jun 10 2025

Formula

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