This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A060834 #53 Apr 17 2025 12:46:13
%S A060834 31,43,67,103,151,211,283,367,463,571,691,823,967,1123,1291,1471,1663,
%T A060834 1867,2083,2311,2551,2803,3067,3343,3631,3931,4243,4567,4903,5251,
%U A060834 5611,5983,6367,6763,7171,7591,8023,8467,8923,9391,9871,10363,10867,11383
%N A060834 a(n) = 6*n^2 + 6*n + 31.
%C A060834 First 29 values are primes.
%C A060834 From _Peter Bala_, Apr 18 2018: (Start)
%C A060834 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.
%C A060834 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).
%C A060834 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)
%C A060834 Also, numbers k such that 2*k/3 - 2/3 - 19 is a perfect square. - _Bruno Berselli_, Apr 23 2018
%C A060834 Equivalently, numbers k such that 6*k - 177 is a square. - _Vincenzo Librandi_, Apr 23 2018
%D A060834 Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 145.
%D A060834 Donald D. Spencer, Computers in Number Theory, Computer Science Press, Rockville, MD, 1982, pp. 118-119.
%H A060834 Harry J. Smith, <a href="/A060834/b060834.txt">Table of n, a(n) for n = 0..1000</a>
%H A060834 T. Piezas, <a href="https://sites.google.com/view/tpiezas/0023-part-2-prime-generating-polynomials">A collection of algebraic identities. 0023: Part 2, Prime Generating Polynomials, Section IV</a>.
%H A060834 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Prime-GeneratingPolynomial.html">Prime-Generating Polynomial</a>.
%H A060834 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A060834 From _R. J. Mathar_, Feb 05 2008: (Start)
%F A060834 O.g.f.: -(31-50*x+31*x^2)/(-1+x)^3.
%F A060834 a(n) = A049598(n)+31. (End)
%F A060834 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
%F A060834 E.g.f.: exp(x)*(31 + 12*x + 6*x^2). - _Stefano Spezia_, Dec 26 2024
%e A060834 a(29)=4903, prime. a(30)=5251, nonprime.
%t A060834 Table[6n^2+6n+31,{n,0,60}] (* or *) LinearRecurrence[{3,-3,1},{31,43,67},60] (* _Harvey P. Dale_, Aug 09 2011 *)
%o A060834 (PARI) a(n) = { 6*n^2 + 6*n + 31 } \\ _Harry J. Smith_, Jul 19 2009
%o A060834 (GAP) List([0..80],n->6*n^2+6*n+31); # _Muniru A Asiru_, Apr 22 2018
%Y A060834 Cf. A049598, A060844, A005846.
%K A060834 nonn,easy
%O A060834 0,1
%A A060834 _Jason Earls_, May 02 2001
%E A060834 More terms from Larry Reeves (larryr(AT)acm.org), May 07 2001