A005471 Primes of the form m^2 + 3m + 9, where m can be positive or negative.

7, 13, 19, 37, 79, 97, 139, 163, 313, 349, 607, 709, 877, 937, 1063, 1129, 1489, 1567, 1987, 2557, 2659, 3313, 3547, 4297, 5119, 5557, 7489, 8017, 8563, 9127, 9319, 9907, 10513, 11779, 12889, 15013, 15259, 16519, 17299, 18097, 18367, 18913, 20029

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Primes of the form m^2 + m + 7, for some m >= 0. - Daniel Forgues, Jan 26 2020

Primes p such that 4*p - 27 is a square. Also, primes p such that the Galois group of the polynomial X^3 - p*X + p over Q is the cyclic group of order 3. See Conrad, Corollary 2.5. - Peter Bala, Oct 17 2021

From Peter Bala, Nov 18 2021: (Start)

Primes p such that the Galois group of the cubic X^3 + p*(X + 1)^2 over Q is the cyclic group C_3.

If p = m^2 + 3*m + 9 is prime then the Galois group of the cubic X^3 - m*X^2 - (m + 3)*X - 1 over Q is C_3. See Shanks.

The pair of cubics X^3 - m*p*X^2 - 3*(m+1)*p*X - (2*m+3)*p and X^3 - 2*p*X^2 + p*(p - 10)*X + p*(p - 8) also have their Galois groups over Q equal to C_3 (both cubics are irreducible over Q by Eisenstein's criteria). Apply Conrad, Corollary 2.5. (End)

Examples

			For m = -11, -10, ..., 22 the primes of the form m^2+3m+9 are 97, 79, 37, 19, 13, 7, 7, 13, 19, 37, 79, 97, 139, 163, 313, 349.
		

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000
• S. Barbero, U. Cerruti, and N. Murru, Identities Involving Zeros of Ramanujan and Shanks Cubic Polynomials, J. Integer Seq., Vol. 16 (2013), Article 13.8.1.
• Keith Conrad, Galois groups of cubics and quartics (not in characteristic 2)
• Hyun Kwang Kim and Jung Soo Kim, Evaluation of zeta function of the simplest cubic field at negative odd integers, Math. Comp. 71 (2002), no. 239, 1243-1262.
• D. Shanks, The simplest cubic fields, Math. Comp., 28 (1974), 1137-1152.
• Koji Uchida, Class numbers of cubic cyclic fields, J. Math. Soc. Japan, Vol. 26, No. 3, 1974, pp. 447 - 453.

Crossrefs

Primes in A027692.

Cf. A175282, A227622, A349461.

Programs

Magma

[a: n in [-1..150] | IsPrime(a) where a is  n^2+3*n+9]; // Vincenzo Librandi, Mar 22 2013

Maple

A005471 := proc(n)
    if n = 1 then
        7;
    else
        A175282(n-1)*(3+A175282(n-1))+9 ;
    end if;
end proc: # R. J. Mathar, Jun 06 2019

Mathematica

Select[Table[n^2 + 3*n + 9, {n, -1, 200}], PrimeQ] (* T. D. Noe, Mar 21 2013 *)

Formula

a(n) == 1 (mod 6). - Zak Seidov, Mar 20 2010

a(n+1) = A175282(n)^2 + 3*A175282(n) + 9. - R. J. Mathar, Jun 06 2019