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

Author

N. J. A. Sloane, Nov 05 2011

Keywords

• nonn
• easy

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

Crossrefs

Cf. A115104, A001912, A002496, A005574.

Programs

Magma

[ a: n in [0..200] | IsPrime(a) where a is 4*n^3+1 ]; // Vincenzo Librandi, Nov 08 2011

Mathematica

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

PARI

for(n=1,1e3,if(isprime(t=4*n^3+1),print1(t", "))) \\ Charles R Greathouse IV, Nov 21 2011