A111251 Numbers k such that 3*k^2 + 3*k + 1 is prime.

1, 2, 3, 4, 6, 9, 10, 11, 13, 14, 17, 23, 24, 25, 27, 28, 30, 32, 34, 37, 38, 41, 42, 45, 48, 49, 52, 55, 58, 62, 63, 66, 67, 74, 80, 81, 86, 88, 90, 91, 93, 95, 105, 108, 119, 123, 125, 128, 129, 136, 140, 142, 147, 153, 156, 157, 158, 164, 165, 170, 171, 172, 175

Offset: 1

Author

Parthasarathy Nambi, Oct 31 2005

Keywords

• nonn

Comments

That is, positive integers k such that (k+1)^3 - k^3 is prime.

The Hardy-Littlewood constant 1.68109913... of this polynomial is approximately half that of the well-known Euler polynomial A221712, i.e., in comparison, only about half as many prime numbers are produced asymptotically as with k^2 + k + 41. - Hugo Pfoertner, Feb 10 2020

The primes that are obtained are called cuban primes and are in A002407. - Bernard Schott, Feb 13 2020

Examples

			For k=52, 3*52^2 + 3*52 + 1 = 8269 is prime, so 52 is a term.
		

Links

• Daniel Starodubtsev, Table of n, a(n) for n = 1..10000 (terms 1..1965 from Pierre CAMI)

Crossrefs

Cf. A221712, A002407 (resulting primes), A002504, A121259.

Programs

Magma

[k: k in [1..180] | IsPrime(3*k^2 + 3*k + 1)]; // Marius A. Burtea, Feb 10 2020

Mathematica

Select[Range[200],PrimeQ[3#^2+3#+1]&] (* Harvey P. Dale, May 29 2017 *)

PARI

for(n=0,250,if(isprime(3*n^2+3*n+1),print1(n,",")))

Formula

a(n) = floor(sqrt(A002407(n)/3)). - Rémi Guillaume, Oct 16 2023

a(n) = A002504(n) - 1. - Rémi Guillaume, Oct 21 2023

a(n) = (A121259(n) - 1)/2. - Rémi Guillaume, Dec 29 2023

Extensions

Extended by Lambert Klasen (lambert.klasen(AT)gmx.net), Nov 02 2005