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 A056899 #42 Feb 16 2025 08:32:43
%S A056899 2,3,11,83,227,443,1091,1523,2027,3251,6563,9803,11027,12323,13691,
%T A056899 15131,21611,29243,47963,50627,56171,59051,62003,65027,74531,88211,
%U A056899 91811,95483,103043,119027,123203,131771,136163,140627,149771,173891
%N A056899 Primes of the form k^2 + 2.
%C A056899 Also, primes of the form k^2 - 2k + 3.
%C A056899 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 A056899 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 A056899 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 A056899 M. Cerasoli, F. Eugeni and M. Protasi, Elementi di Matematica Discreta, Bologna 1988.
%D A056899 Emanuele Munarini and Norma Zagaglia Salvi, Matematica Discreta, UTET, CittaStudiEdizioni, Milano 1997.
%H A056899 T. D. Noe, <a href="/A056899/b056899.txt">Table of n, a(n) for n = 1..1000</a>
%H A056899 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Near-SquarePrime.html">Near-Square Prime</a>
%F A056899 For n>1, a(n) = 72*A000217(A056900(n-2))+11
%F A056899 a(n) = A067201(n)^2 + 2. - _M. F. Hasler_, Apr 05 2009
%p A056899 select(isprime, [seq(t^2+2, t = 0..1000)]); # _Robert Israel_, Sep 03 2015
%t A056899 Select[ Range[0, 500]^2 + 2, PrimeQ] (* _Robert G. Wilson v_, Sep 03 2015 *)
%o A056899 (Magma) [n: n in PrimesUpTo(175000) | IsSquare(n-2)]; // _Bruno Berselli_, Apr 05 2011
%o A056899 (Magma) [ a: n in [0..450] | IsPrime(a) where a is n^2 +2 ]; // _Vincenzo Librandi_, Apr 06 2011
%o A056899 (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 A056899 Intersection of A146327 and A000040; intersection of A059100 and A000040.
%Y A056899 Cf. A002496.
%K A056899 nonn
%O A056899 1,1
%A A056899 _Henry Bottomley_, Jul 05 2000