OFFSET
0,1
COMMENTS
First 29 values are primes.
From Peter Bala, Apr 18 2018: (Start)
Let P(n) = 6*n^2 + 6*n + 31. The polynomial P(2*n-14) = 24*n^2 - 660*n + 4567 takes distinct prime values for n = 0 to 28.
The value of the polynomial 2*P(3/2*(n-10)) = 27*n^2 - 522*n + 2582 for n = 0 to 22 is either double a prime or a prime (alternately).
The value of the polynomial 4*P(4/3*(n-9)) = 32*n^2 - 552*n + 2469 for n = 0 to 28 is either prime or 3 times a prime, except when n = 16. (End)
Also, numbers k such that 2*k/3 - 2/3 - 19 is a perfect square. - Bruno Berselli, Apr 23 2018
Equivalently, numbers k such that 6*k - 177 is a square. - Vincenzo Librandi, Apr 23 2018
REFERENCES
Donald D. Spencer, Computers in Number Theory, Computer Science Press, Rockville, MD, 1982, pp. 118-119.
LINKS
Harry J. Smith, Table of n, a(n) for n = 0..1000
T. Piezas, A collection of algebraic identities. 0023: Part 2, Prime Generating Polynomials, Section IV.
Eric Weisstein's World of Mathematics, Prime-Generating Polynomial.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
FORMULA
From R. J. Mathar, Feb 05 2008: (Start)
O.g.f.: -(31-50*x+31*x^2)/(-1+x)^3.
a(n) = A049598(n)+31. (End)
a(0)=31, a(1)=43, a(2)=67, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, Aug 09 2011
E.g.f.: exp(x)*(31 + 12*x + 6*x^2). - Stefano Spezia, Dec 26 2024
EXAMPLE
a(29)=4903, prime. a(30)=5251, nonprime.
MATHEMATICA
Table[6n^2+6n+31, {n, 0, 60}] (* or *) LinearRecurrence[{3, -3, 1}, {31, 43, 67}, 60] (* Harvey P. Dale, Aug 09 2011 *)
PROG
(PARI) a(n) = { 6*n^2 + 6*n + 31 } \\ Harry J. Smith, Jul 19 2009
(GAP) List([0..80], n->6*n^2+6*n+31); # Muniru A Asiru, Apr 22 2018
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Jason Earls, May 02 2001
EXTENSIONS
More terms from Larry Reeves (larryr(AT)acm.org), May 07 2001
STATUS
approved