A384603 -id:A384603 - cp's OEIS frontend

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

Alain Rocchelli, Jun 04 2025

nonn

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.

			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.

Cf. A384603.

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, ", ")));

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

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

A385093 Primes preceded and followed by gaps whose quotient (value greater or equal to 1) is greater than 2 and less than 3.

Original entry on oeis.org

331, 953, 1381, 1861, 2161, 2357, 2371, 2423, 2879, 3229, 3271, 3407, 3491, 3607, 3643, 3691, 3889, 4057, 4073, 4139, 4201, 4507, 4567, 4751, 4831, 4903, 4987, 5059, 5153, 5297, 5309, 5683, 5897, 6029, 6053, 6067, 6173, 6229, 6529, 6599, 6653, 6857, 7079, 7151, 7159, 7193, 7283, 7417, 7717, 7867, 7963

Offset: 1

Alain Rocchelli, Jun 17 2025

nonn

Primes prime(k) such that 2 < Max(prime(k)-prime(k-1), prime(k+1)-prime(k)) / Min(prime(k)-prime(k-1), prime(k+1)-prime(k)) < 3.

			331 is a term because Max(331-317,337-331)/Min(331-317,337-331) = 14/6 = 2.3333.
953 is a term because Max(953-947,967-953)/Min(953-947,967-953) = 14/6 = 2.3333.
1381 is a term because Max(1381-1373,1399-1381)/Min(1381-1373,1399-1381) = 18/8 = 2.25.

Cf. A384603, A384618.

Prime/@Select[Range[3,1000],2James C. McMahon, Jun 29 2025 *)

forprime(P=3, 8000, my(M=P-precprime(P-1), Q=nextprime(P+1)-P, AR=max(M,Q)/min(M,Q)); if(AR>2 && AR<3, print1(P,", ")));

Conjecture: Limit_{n->oo} n / PrimePi(a(n)) = 1/6.

cp's OEIS Frontend

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

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