A356475 First of three consecutive primes p,q,r such that p*q + q*r + r*p is prime.

2, 3, 5, 7, 17, 29, 37, 41, 43, 67, 83, 103, 137, 157, 179, 181, 193, 227, 277, 283, 347, 359, 383, 431, 457, 461, 607, 661, 701, 709, 757, 773, 823, 827, 839, 859, 937, 967, 1013, 1051, 1061, 1109, 1129, 1187, 1201, 1213, 1249, 1283, 1307, 1327, 1373, 1423, 1439, 1471, 1481, 1487, 1543, 1567

Offset: 1

Author

J. M. Bergot and Robert Israel, Aug 08 2022

Keywords

• nonn

Examples

			a(4) = 7 is a term because 7, 11, 13 are three consecutive primes with 7*11 + 11*13 + 13*7 = 311 which is prime.
		

Links

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

Crossrefs

Cf. A189759, A356471, A356477.

Programs

Maple

R:= NULL: count:= 0:
P:= Vector(3,ithprime):
while count < 100 do
  x:= P[1]*P[2]+P[2]*P[3]+P[3]*P[1];
  if isprime(x) then R:= R, P[1]; count:= count+1 fi;
  P[1..2]:= P[2..3];
  P[3]:= nextprime(P[3]);
od:
R;

Mathematica

Select[Partition[Prime[Range[250]], 3, 1], PrimeQ[Total[# * RotateLeft[#]]] &][[;; , 1]] (* Amiram Eldar, Aug 08 2022 *)

PARI

list(lim)=my(v=List(),p=2,q=3); forprime(r=5,nextprime(nextprime(lim\1+1)+1), if(isprime(p*q + q*r + r*p), listput(v,p)); p=q; q=r); Vec(v) \\ Charles R Greathouse IV, Sep 06 2022

Python

from itertools import islice
from sympy import isprime, nextprime
def agen():
    p, q, r = 2, 3, 5
    while True:
        if isprime(p*q + q*r + r*p): yield p
        p, q, r = q, r, nextprime(r)
print(list(islice(agen(), 58))) # Michael S. Branicky, Aug 08 2022