A386540 Primes p such that 2p - 1, 3p - 2, (p + 1)/2, and (p + 2)/3 are also prime.

37, 2557, 3061, 5581, 88741, 124021, 157081, 178537, 216217, 216757, 217057, 330661, 344821, 352081, 387577, 423481, 459397, 477577, 521137, 790861, 806521, 865957, 869521, 1369657, 1517881, 1673401, 1704397, 1710661, 1970257, 2132797, 2292781, 2361781, 2680141

Offset

1,1

Comments

All terms are congruent to 1 modulo 12.

Let q = (p + 1)/2 and r = (p + 2)/3; then 3r = 2q + 1.

Links

Robert Israel, Table of n, a(n) for n = 1..1000

Example

37 is a term, since it is prime and 2*37 - 1 = 73, 3*37 - 2 = 109, (37 + 1)/2 = 19 and (37 + 2)/3 = 13 are all prime.

Maple

select(p -> andmap(isprime, [p, 2*p-1, 3*p-2, (p+1)/2, (p+2)/3]), [seq(1+12*i, i=1..10^6)]); # Robert Israel, Jul 25 2025

Mathematica

Select[Prime[Range[2*10^5]], AllTrue[{2#-1, 3#-2, (#+1)/2, (#+2)/3}, PrimeQ]&] (* James C. McMahon, Jul 25 2025 *)

Prog

(Python)
from gmpy2 import is_prime
def ok(p): return p&1 and p%3 == 1 and all(is_prime(q) for q in [p, 2*p-1, 3*p-2, (p+1)//2, (p+2)//3])
print([k for k in range(1, 10**7, 12) if ok(k)]) # Michael S. Branicky, Jul 25 2025

Crossrefs

Intersection of A036570 and A174734.

Sequence in context: A099807 A193123 A295846 * A297719 A296738 A296385

Adjacent sequences: A386537 A386538 A386539 * A386541 A386542 A386543

Keyword

nonn,easy

Author

Holger Wallenta, Jul 25 2025

Status

approved