A060834 a(n) = 6*n^2 + 6*n + 31.

31, 43, 67, 103, 151, 211, 283, 367, 463, 571, 691, 823, 967, 1123, 1291, 1471, 1663, 1867, 2083, 2311, 2551, 2803, 3067, 3343, 3631, 3931, 4243, 4567, 4903, 5251, 5611, 5983, 6367, 6763, 7171, 7591, 8023, 8467, 8923, 9391, 9871, 10363, 10867, 11383

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

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) { for (n=0, 1000, write("b060834.txt", 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

Cf. A049598, A060844, A005846.

Sequence in context: A118637 A096163 A139883 * A060844 A112789 A266989

Adjacent sequences: A060831 A060832 A060833 * A060835 A060836 A060837

Keyword

nonn,easy

Author

Jason Earls, May 02 2001

Extensions

More terms from Larry Reeves (larryr(AT)acm.org), May 07 2001

Status

approved