A199307 Primes of the form 4n^3 + 1.

5, 109, 257, 1373, 2917, 4001, 27437, 62501, 157217, 202613, 237277, 296353, 470597, 629857, 665501, 1492993, 1556069, 1898209, 2456501, 2634013, 3217429, 3322337, 4244833, 5038849, 5180117, 6572129, 10512289, 11453153, 12706093

Offset

1,1

Comments

Dirichlet's theorem on primes in arithmetic progressions tells us, for example, that there are infinitely many primes of the form 4n+1. For primes represented by polynomials of degree greater than 1, the Bateman-Horn paper gives a conjecture on the density.

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

P. Bateman and R. A. Horn, A heuristic asymptotic formula concerning the distribution of prime numbers, Mathematics of Computation, 16 (1962), 363-367.

Wikipedia, Bateman-Horn Conjecture

Mathematica

Select[Table[4n^3+1, {n, 1100}], PrimeQ] (* Vincenzo Librandi, Aug 01 2012 *)

Prog

(Magma) [ a: n in [0..200] | IsPrime(a) where a is 4*n^3+1 ]; // Vincenzo Librandi, Nov 08 2011
(PARI) for(n=1, 1e3, if(isprime(t=4*n^3+1), print1(t", "))) \\ Charles R Greathouse IV, Nov 21 2011

Crossrefs

Cf. A115104, A001912, A002496, A005574.

Sequence in context: A305095 A128110 A316755 * A195561 A142510 A195552

Adjacent sequences: A199304 A199305 A199306 * A199308 A199309 A199310

Keyword

nonn,easy

Author

N. J. A. Sloane, Nov 05 2011

Status

approved