A007641 - OEIS

%I M5219 #44 Dec 04 2024 10:34:08

%S 29,31,37,47,61,79,101,127,157,191,229,271,317,367,421,479,541,607,

%T 677,751,829,911,997,1087,1181,1279,1381,1487,1597,1951,2207,2341,

%U 2621,2767,2917,3229,3391,3557,3727,4079,4261,4447,4637,4831,5231,5437

%N Primes of the form 2*k^2 + 29.

%C The first 29 terms of 2*k^2 + 29 (k = 0 to 28) are primes. This was discovered by Adrien-Marie Legendre. The sequence and its first 8 terms appear in the novel Code to Zero by Ken Follett. - _Amiram Eldar_, Apr 08 2017

%C Let P(k) = 2*k^2 + 29. The polynomial P(2*k - 28) = 8*k^2 - 224*k + 1597 produces prime values (not distinct) for k = 0 to 28. The polynomial P(3*k - 55) = 18*k^2 - 660*k + 6079 produces distinct prime values for k = 0 to 27. Cf. A050265. - _Peter Bala_, Apr 16 2018

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%D Ken Follett, Code to Zero, New York: Signet, 2001, p. 18.

%H Vincenzo Librandi, <a href="/A007641/b007641.txt">Table of n, a(n) for n = 1..1000</a>

%H Adrien-Marie Legendre, <a href="http://archive.org/details/essaisurlathor00lege">Essai sur la théorie des nombres</a>, Paris: Duprat, 1798, p. 10.

%H Jitender Singh, <a href="https://arxiv.org/abs/2411.18366">Prime numbers and factorization of polynomials</a>, arXiv:2411.18366 [math.NT], 2024.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Prime-GeneratingPolynomial.html">Prime-generating Polynomial.</a>

%t Select[Table[2 n^2 + 29, {n, 0, 70}], PrimeQ] (* _Vincenzo Librandi_, Mar 20 2013 *)

%o (Magma) [a: n in [0..60] | IsPrime(a) where a is 2*n^2+29]; // _Vincenzo Librandi_, Mar 20 2013

%o (PARI) list(lim)=my(v=List(),t); for(n=0,sqrtint((lim-29)\2), if(isprime(t=2*n^2+29), listput(v,t))); Vec(v) \\ _Charles R Greathouse IV_, Jan 20 2022

%Y Cf. A005846, A050265, A352800.

%K nonn,easy

%O 1,1

%A _N. J. A. Sloane_, _Mira Bernstein_, _Robert G. Wilson v_

%E Edited by _Erich Friedman_, Feb 09 2002