|
%I M4345 #74 Sep 08 2022 08:44:33
%S 7,13,19,37,79,97,139,163,313,349,607,709,877,937,1063,1129,1489,1567,
%T 1987,2557,2659,3313,3547,4297,5119,5557,7489,8017,8563,9127,9319,
%U 9907,10513,11779,12889,15013,15259,16519,17299,18097,18367,18913,20029
%N Primes of the form m^2 + 3m + 9, where m can be positive or negative.
%C Primes of the form m^2 + m + 7, for some m >= 0. - _Daniel Forgues_, Jan 26 2020
%C 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
%C From _Peter Bala_, Nov 18 2021: (Start)
%C Primes p such that the Galois group of the cubic X^3 + p*(X + 1)^2 over Q is the cyclic group C_3.
%C 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.
%C 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)
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A005471/b005471.txt">Table of n, a(n) for n = 1..1000</a>
%H S. Barbero, U. Cerruti, and N. Murru, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Barbero/barbero11.html">Identities Involving Zeros of Ramanujan and Shanks Cubic Polynomials</a>, J. Integer Seq., Vol. 16 (2013), Article 13.8.1.
%H Keith Conrad, <a href="http://www.ms.uky.edu/~sohum/ma561/notes/workspace/books/cubicquartic.pdf">Galois groups of cubics and quartics (not in characteristic 2)</a>
%H Hyun Kwang Kim and Jung Soo Kim, <a href="http://dx.doi.org/10.1090/S0025-5718-02-01395-9">Evaluation of zeta function of the simplest cubic field at negative odd integers</a>, Math. Comp. 71 (2002), no. 239, 1243-1262.
%H D. Shanks, <a href="http://dx.doi.org/10.1090/S0025-5718-1974-0352049-8">The simplest cubic fields</a>, Math. Comp., 28 (1974), 1137-1152.
%H Koji Uchida, <a href="https://doi:10.2969/jmsj/02630447">Class numbers of cubic cyclic fields</a>, J. Math. Soc. Japan, Vol. 26, No. 3, 1974, pp. 447 - 453.
%F a(n) == 1 (mod 6). - _Zak Seidov_, Mar 20 2010
%F a(n+1) = A175282(n)^2 + 3*A175282(n) + 9. - _R. J. Mathar_, Jun 06 2019
%e 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.
%p A005471 := proc(n)
%p if n = 1 then
%p 7;
%p else
%p A175282(n-1)*(3+A175282(n-1))+9 ;
%p end if;
%p end proc: # _R. J. Mathar_, Jun 06 2019
%t Select[Table[n^2 + 3*n + 9, {n, -1, 200}], PrimeQ] (* _T. D. Noe_, Mar 21 2013 *)
%o (Magma) [a: n in [-1..150] | IsPrime(a) where a is n^2+3*n+9]; // _Vincenzo Librandi_, Mar 22 2013
%Y Primes in A027692.
%Y Cf. A175282, A227622, A349461.
%K nonn,easy
%O 1,1
%A _N. J. A. Sloane_
|
