OFFSET
0,1
COMMENTS
This quadratic seems to be good at generating many distinct primes. It generates 642 primes for n < 10^3, 5132 primes for n < 10^4, 41224 primes for n < 10^5 and 340009 primes for n < 10^6. The first few primes generated are 1697, 4027, 6359, 8693, 13367, 18049, 20393, 22739, 25087, 27437.
The quadratic was first discovered as b(n) = n^2 + 1151n - 1023163 by Steve Trevorrow in 2006 during Al Zimmermann's Prime Generating Polynomials contest (see link). Note that a(n) = b(n + 589).
Smallest n such that a(n) is not squarefree is 704; a(704) = 2136929 = 73^2*401 and smallest n such that a(n) is square is 1327; a(1327) = 4853209 = 2203^2. - Altug Alkan, Mar 30 2018
LINKS
Colin Barker, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Prime-Generating Polynomial
Al Zimmermann, Prime Generating Polynomials contest, 2006.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
FORMULA
a(n) = 2a(n-1) - a(n-2) + 2, a(1) = 1697, a(2) = 4027.
From Colin Barker, Mar 30 2018: (Start)
G.f.: (1697 - 1064*x - 631*x^2) / (1 - x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>2.
(End)
MATHEMATICA
Table[n^2 + 2329 n + 1697, {n, 0, 50}] (* Vincenzo Librandi, Mar 31 2018 *)
PROG
(PARI) a(n) = n^2 + 2329*n + 1697; \\ Altug Alkan, Mar 30 2018
(PARI) Vec((1697 - 1064*x - 631*x^2) / (1 - x)^3 + O(x^60)) \\ Colin Barker, Mar 30 2018
(Magma) [n^2+2329*n+1697: n in [0..50]]; // Vincenzo Librandi, Mar 31 2018
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Dmitry Kamenetsky, Mar 30 2018
STATUS
approved