A384603 Primes preceded and followed by gaps whose quotient (value greater or equal to 1) is less than 2.

5, 23, 37, 47, 53, 67, 79, 83, 89, 131, 157, 163, 167, 173, 211, 233, 251, 257, 263, 277, 293, 337, 353, 359, 367, 373, 379, 383, 389, 409, 439, 443, 449, 479, 503, 547, 557, 563, 577, 587, 593, 607, 613, 631, 647, 653, 677, 683, 691, 701, 709, 719, 727, 733, 739, 751, 757, 787, 797

Offset

1,1

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.

Links

Paolo Xausa, Table of n, a(n) for n = 1..10000

Formula

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

Example

5 is a term because Max(5-3,7-5)/Min(5-3,7-5) = 2/2 = 1.
23 is a term because Max(23-19,29-23)/Min(23-19,29-23) = 6/4 = 1.5.
37 is a term because Max(37-31,41-37)/Min(37-31,41-37) = 6/4 = 1.5.

Mathematica

Select[Partition[Prime[Range[200]], 3, 1], Max[#] < 2*Min[#] & [Differences[#]] &][[All, 2]] (* Paolo Xausa, Jan 21 2026 *)

Prog

(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 A384603_gen(): # generator of terms
    p, q, r = 2, 3, 5
    while True:
        s, t = q-p, r-q
        if s<(t<<1) and t<(s<<1): yield q
        p, q, r = q, r, nextprime(r)
A384603_list = list(islice(A384603_gen(), 59)) # Chai Wah Wu, Jun 10 2025

Crossrefs

Cf. A383215, A384618.

Sequence in context: A337436 A050906 A195974 * A098421 A371622 A044447

Adjacent sequences: A384600 A384601 A384602 * A384604 A384605 A384606

Keyword

nonn

Author

Alain Rocchelli, Jun 04 2025

Status

approved