A358149 First of four consecutive primes p,q,r,s such that (2*p+q)/5 and (r+2*s)/5 are prime.

11, 1151, 33071, 33637, 55331, 57637, 75997, 90821, 97007, 100151, 112237, 118219, 123581, 141629, 154459, 160553, 165961, 199247, 212777, 222823, 288361, 289511, 293677, 319993, 329471, 331697, 336101, 361799, 364537, 375371, 381467, 437279, 437693, 442571, 444461, 457837, 475751, 490877, 540781

Offset: 1

Author

J. M. Bergot and Robert Israel, Nov 01 2022

Keywords

• nonn

Comments

Dickson's conjecture implies there are infinitely many terms where q = p+2, r = p+6 and s = p+8; the first two of these are 11 and 55331.

Examples

			a(3) = 33071 is a term because 33071, 33073, 33083, 33091 are four consecutive primes with (2*33071+33073)/5 = 19843 and (33083+2*33091)/5 = 19853 prime.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A358155.

Programs

Maple

Res:= NULL: count:= 0:
q:= 2: r:= 3: s:= 5:
while count < 50 do
  p:= q; q:= r; r:= s; s:= nextprime(s);
  t:= (2*p+q)/5; u:= (r+2*s)/5;
  if (t::integer and u::integer and isprime(t) and isprime(u))
   then
    count:= count+1; Res:= Res,p;
  fi
od:
Res;

Mathematica

Select[Partition[Prime[Range[45000]], 4, 1], PrimeQ[(2*#[[1]] + #[[2]])/5] && PrimeQ[(#[[3]] + 2*#[[4]])/5] &][[;; , 1]] (* Amiram Eldar, Nov 01 2022 *)