A300845 a(n) is the smallest prime q such that q^2 + q*p + p^2 is a prime number where p is n-th prime, or 0 if no such q exists.

3, 2, 7, 2, 3, 2, 3, 11, 3, 3, 3, 2, 7, 3, 19, 7, 7, 2, 11, 13, 2, 5, 37, 19, 11, 3, 5, 3, 5, 13, 3, 7, 7, 2, 7, 5, 2, 3, 37, 7, 3, 29, 13, 5, 3, 11, 17, 29, 37, 2, 13, 3, 2, 67, 19, 7, 7, 5, 3, 3, 29, 43, 23, 7, 5, 3, 3, 5, 7, 2, 43, 3, 2, 17, 17, 7, 19, 2, 13, 23, 43, 3, 7, 2, 2, 7, 7, 2, 7

Offset: 1

Author

Altug Alkan, Mar 13 2018

Keywords

• nonn

Comments

Probably, for each prime p, there is prime q such that q^2 + q*p + p^2 is also a prime since the existence of such q is a special case of Hypothesis H of Schinzel and Sierpinski. However, this is not proven yet.

Corresponding generalized cuban primes are 19, 19, 109, 67, 163, 199, 349, 691, 607, 937, 1063, 1447, 2017, 1987, 3463, 3229, 3943, 3847, 5347, 6133, ...

Examples

			a(3) = 7 because 7^2 + 7*5 + 5^2 = 109 is prime number and 7 is the least prime with this property.
		

Links

• Wikipedia, Schinzel's Hypothesis H

Crossrefs

Cf. A007645, A244146, A244175.

Programs

Maple

f:= proc(p) local q;
  q:= 1;
  do
    q:= nextprime(q);
    if isprime(q^2+q*p+p^2) then return q fi;
  od
end proc:
map(f, select(isprime, [2,seq(i,i=3..1000,2)])); # Robert Israel, Mar 13 2018

Mathematica

Table[Block[{q = 2}, While[! PrimeQ[q^2 + q p + p^2], q = NextPrime@ q]; q], {p, Prime@ Range@ 89}] (* Michael De Vlieger, Mar 14 2018 *)

PARI

a(n) = {my(p=prime(n)); forprime(q=2, ,if(isprime(p^2+p*q+q^2), return(q)))}