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

Author

Jason Earls, May 02 2001

Keywords

• nonn
• easy

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

Examples

			a(29)=4903, prime. a(30)=5251, nonprime.
		

References

• Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 145.
• 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).

Crossrefs

Cf. A049598, A060844, A005846.

Programs

GAP

List([0..80],n->6*n^2+6*n+31); # Muniru A Asiru, Apr 22 2018

Mathematica

Table[6n^2+6n+31,{n,0,60}] (* or *) LinearRecurrence[{3,-3,1},{31,43,67},60] (* Harvey P. Dale, Aug 09 2011 *)

PARI

a(n) = { 6*n^2 + 6*n + 31 } \\ Harry J. Smith, Jul 19 2009

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

Extensions

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