A187058 Primes p such that the polynomial x^2 + x + p generates only primes for x = 1..5.

11, 17, 41, 1277, 1607, 3527, 13901, 21557, 26681, 28277, 31247, 33617, 55661, 68897, 97367, 113147, 128981, 166841, 195731, 221717, 347981, 348431, 354371, 416387, 421697, 506327, 548831, 566537, 665111, 844427, 929057, 954257

Offset: 1

Author

Jonathan Vos Post, Mar 03 2011

Keywords

• nonn

Comments

From Weber, p. 15.

This sequence is infinite, assuming Dickson's conjecture.

All terms = {11, 17} mod 30. - Zak Seidov, May 07 2011

Examples

			a(2) = 17 because x^2 + x + 17 generates 0^2 + 0 + 17 = 17; 1^2 + 1 + 17 = 19; 2^2 + 2 + 17 = 23; 3^2 + 3 + 17 = 29; 4^2 + 4 + 17 = 37; and 5^2 + 5 + 17 = 47, all primes.
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000, replacing a b-file from Zak Seidov.
• H. J. Weber, Regularities of Twin, Triplet and Multiplet Prime Numbers, arXiv:1103.0447 [math.NT], 2011-2012.

Crossrefs

Cf. A144051, A187057, A187060.

Programs

Mathematica

okQ[n_] := And @@ PrimeQ[Table[i^2 + i + n, {i, 0, 5}]]; Select[Range[10000], okQ] (* T. D. Noe, Mar 03 2011 *)
Select[Prime[Range[76000]],AllTrue[#+{2,6,12,20,30},PrimeQ]&] (* Harvey P. Dale, Apr 12 2025 *)

PARI

forprime(p=9,1e6,if((p%30==11 || p%30==17) && isprime(p+2) && isprime(p+6) && isprime(p+12) && isprime(p+20) && isprime(p+30), print1(p", "))) \\ Charles R Greathouse IV, May 08 2011