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A005471
Primes of the form m^2 + 3m + 9, where m can be positive or negative.
29
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
1,1
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)
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
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.
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.
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
EXAMPLE
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.
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 *)
PROG
(Magma) [a: n in [-1..150] | IsPrime(a) where a is n^2+3*n+9]; // Vincenzo Librandi, Mar 22 2013
CROSSREFS
Primes in A027692.
Sequence in context: A266268 A110074 A058383 * A249381 A040096 A358322
KEYWORD
nonn,easy
STATUS
approved