A187057 Primes p such that the polynomial x^2 + x + p generates only primes for x = 0, ..., 4.
11, 17, 41, 347, 641, 1277, 1427, 1607, 2687, 3527, 4001, 4637, 4931, 13901, 19421, 21011, 21557, 22271, 23741, 26681, 26711, 27941, 28277, 31247, 32057, 33617, 43781, 45821, 55331, 55661, 55817, 68207, 68897, 71327, 91571, 97367, 113147, 128657, 128981
Offset: 1
Keywords
Examples
a(1) = 11 because x^2 + x + 11 generates 0^2 + 0 + 11; 1^2 + 1 + 11 = 13; 2^2 + 2 + 11 = 17; 3^2 + 3 + 11 = 23; 4^2 + 4 + 11 = 31, all primes.
Links
- Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
- H. J. Weber, Regularities of Twin, Triplet and Multiplet Prime Numbers, Mar 2, 2011.
Programs
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Maple
q:= p-> andmap(x-> isprime(x^2+x+p), [$0..4]): select(q, [i*6+5$i=0..25000])[]; # Alois P. Heinz, Jun 16 2025
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Mathematica
okQ[n_] := And @@ PrimeQ[Table[i^2 + i + n, {i, 0, 4}]]; Select[Range[10000], okQ] (* T. D. Noe, Mar 03 2011 *) Select[Prime[Range[12500]],AllTrue[#+{2,6,12,20},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Aug 11 2019 *) -
PARI
forprime(p=2, 1e4, if(isprime(p+2)&&isprime(p+6)&&isprime(p+12) &&isprime(p+20), print1(p", "))) \\ Charles R Greathouse IV, Mar 04 2012
Comments