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

1,1

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, ...

Links

Table of n, a(n) for n=1..89.

Wikipedia, Schinzel's Hypothesis H

Example

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

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 *)

Prog

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

Crossrefs

Cf. A007645, A244146, A244175.

Sequence in context: A326607 A071189 A137822 * A344766 A302714 A193574

Adjacent sequences: A300842 A300843 A300844 * A300846 A300847 A300848

Keyword

nonn

Author

Altug Alkan, Mar 13 2018

Status

approved