Search: id:A056899
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%I A056899
%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 n^2+2.
%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=6m+3 
               then a(n)=k^2+2=72m(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 E. (labos(AT)ana.sote.hu), 
               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). [From M. F. Hasler (MHasler(AT)univ-ag.fr), 
               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 Eric Weisstein's World of Mathematics, <a href="http://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 Also, primes of form n^2 - 2n + 3.
%F A056899 a(n)=A067201(n)^2+2. [From M. F. Hasler (MHasler(AT)univ-ag.fr), Apr 
               05 2009]
%t A056899 Intersection[Table[n^2+2,{n,0,10^2}],Prime[Range[9*10^3]]] ...or... For[i=2,
               i<=2,a={};Do[If[PrimeQ[n^2+i],AppendTo[a,n^2+i]],{n,0,100}];Print["n^2+",
               i,",",a];i++ ] - Vladimir Orlovsky (4vladimir(AT)gmail.com), Apr 
               29 2008
%Y A056899 Cf. A002496.
%Y A056899 Sequence in context: A008510 A042165 A089921 this_sequence A117699 A065378 
               A161721
%Y A056899 Adjacent sequences: A056896 A056897 A056898 this_sequence A056900 A056901 
               A056902
%K A056899 nonn
%O A056899 1,1
%A A056899 Henry Bottomley (se16(AT)btinternet.com), Jul 05 2000

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