A301985 a(n) = n^2 + 2329*n + 1697.

1697, 4027, 6359, 8693, 11029, 13367, 15707, 18049, 20393, 22739, 25087, 27437, 29789, 32143, 34499, 36857, 39217, 41579, 43943, 46309, 48677, 51047, 53419, 55793, 58169, 60547, 62927, 65309, 67693, 70079, 72467, 74857, 77249, 79643, 82039, 84437, 86837, 89239, 91643, 94049, 96457, 98867, 101279

Offset: 0

Author

Dmitry Kamenetsky, Mar 30 2018

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A005846, A050267, A050268.

Programs

Magma

[n^2+2329*n+1697: n in [0..50]]; // Vincenzo Librandi, Mar 31 2018

Mathematica

Table[n^2 + 2329 n + 1697, {n, 0, 50}] (* Vincenzo Librandi, Mar 31 2018 *)

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

Formula

a(n) = 2*a(n-1) - a(n-2) + 2, a(0) = 1697, a(1) = 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)

E.g.f.: exp(x)*(1697 + 2330*x + x^2). - Elmo R. Oliveira, Feb 10 2025