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%I #41 Sep 08 2022 08:45:01
%S 2,3,11,83,227,443,1091,1523,2027,3251,6563,9803,11027,12323,13691,
%T 15131,21611,29243,47963,50627,56171,59051,62003,65027,74531,88211,
%U 91811,95483,103043,119027,123203,131771,136163,140627,149771,173891
%N Primes of the form k^2 + 2.
%C Also, primes of the form k^2 - 2k + 3.
%C Note that all terms after the first two are equal to 11 modulo 72 and that (a(n)-11)/72 is a triangular number, since they have to be 2 more than the square of an odd multiple of 3 to be prime, and if k = 6*m+3 then a(n) = k^2 + 2 = 72*m*(m+1)/2 + 11.
%C The quotient cycle length is 2 in the continued fraction expansion of sqrt(p) for these primes. E.g.: cfrac(sqrt(6563),6) = 81+1/(81+1/(162+1/(81+1/(162+1/(81+1/(162+`...`)))))). - _Labos Elemer_, Feb 22 2001
%C Primes in A059100; except for a(2)=3 a subsequence of A007491 and congruent to 2 modulo 9. For n>2, a(n)=11 (mod 72). - _M. F. Hasler_, Apr 05 2009
%D M. Cerasoli, F. Eugeni and M. Protasi, Elementi di Matematica Discreta, Bologna 1988.
%D Emanuele Munarini and Norma Zagaglia Salvi, Matematica Discreta, UTET, CittaStudiEdizioni, Milano 1997.
%H T. D. Noe, <a href="/A056899/b056899.txt">Table of n, a(n) for n = 1..1000</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Near-SquarePrime.html">Near-Square Prime</a>
%F For n>1, a(n) = 72*A000217(A056900(n-2))+11
%F a(n) = A067201(n)^2 + 2. - _M. F. Hasler_, Apr 05 2009
%p select(isprime, [seq(t^2+2, t = 0..1000)]); # _Robert Israel_, Sep 03 2015
%t Select[ Range[0, 500]^2 + 2, PrimeQ] (* _Robert G. Wilson v_, Sep 03 2015 *)
%o (Magma) [n: n in PrimesUpTo(175000) | IsSquare(n-2)]; // _Bruno Berselli_, Apr 05 2011
%o (Magma) [ a: n in [0..450] | IsPrime(a) where a is n^2 +2 ]; // _Vincenzo Librandi_, Apr 06 2011
%o (PARI) print1("2, 3");forstep(n=3,1e4,6,if(isprime(t=n^2+2),print1(", "t))) \\ _Charles R Greathouse IV_, Jul 19 2011
%Y Intersection of A146327 and A000040; intersection of A059100 and A000040.
%Y Cf. A002496.
%K nonn
%O 1,1
%A _Henry Bottomley_, Jul 05 2000
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