A240971 Primes p such that (p^2 + p + 1)/3 is prime.

7, 13, 19, 31, 43, 73, 97, 103, 127, 157, 199, 223, 241, 271, 409, 421, 661, 673, 727, 859, 883, 937, 1021, 1039, 1051, 1063, 1093, 1447, 1483, 1609, 1657, 1669, 1723, 1753, 1861, 1879, 1993, 2029, 2203, 2437, 2539, 2677, 2719, 2803, 2833, 2953, 3079, 3121

Offset: 1

Author

Vincenzo Librandi, Aug 05 2014

Keywords

• nonn

Comments

Under Schinzel's hypothesis, there are infinitely many primes of this form.

p must be of form 6k+1 to give an integer. A053182 lists when p^2 + p + 1 is prime. - Jens Kruse Andersen, Aug 06 2014

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..4900
• Eric Weisstein's World of Mathematics, Schinzel's Hypothesis.

Crossrefs

Cf. A053182.

Programs

Magma

[p: p in PrimesInInterval(3,3500)| IsPrime((p^2+p+1) div 3)];

Maple

select(n -> isprime(n) and isprime((n^2 + n + 1)/3), [seq(6*k+1,k=1..1000)]); # Robert Israel, Aug 05 2014

Mathematica

Select[Prime[Range[500]], PrimeQ[(#^2 + # + 1)/3] &]

PARI

forprime(p=1,10^4,s=(p^2+p+1)/3;if(floor(s)==s,if(isprime(s),print1(p,", ")))) \\ Derek Orr, Aug 05 2014