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 A005471 Primes of form n^2 + 3n + 9, where n can be positive or negative. (Formerly M4345) 17

%I M4345

%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 form n^2 + 3n + 9, where n can be positive or negative.

%C Primes of the form n^2 + n + 7, for some n >= 0. - _Daniel Forgues_, Jan 26 2020

%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, 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 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.

%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 n = -11, -10, ..., 22 the primes of the form n^2+3n+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.

%K nonn,easy

%O 1,1

%A _N. J. A. Sloane_.

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Last modified April 17 08:34 EDT 2021. Contains 343064 sequences. (Running on oeis4.)