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

Author

Holger Wallenta, Jul 25 2025

Keywords

• nonn
• easy

Comments

All terms are congruent to 1 modulo 12.

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

Examples

			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.
		

Links

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

Crossrefs

Intersection of A036570 and A174734.

Programs

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 *)

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