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%I #27 Feb 14 2018 04:25:47
%S 3,11,19,41,53,59,73,83,101,107,131,137,149,163,179,181,227,233,251,
%T 293,307,347,389,401,443,467,491,521,523,563,587,593,613,641,677,739,
%U 773,809,811,821,883
%N Primes of the form x^2 + y^2 + 1, x>0, y>0.
%C Sequence is known to be infinite due to a result of Linnik.
%D D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, p. 11.
%H Charles R Greathouse IV, <a href="/A079544/b079544.txt">Table of n, a(n) for n = 1..10000</a>
%H Aminu Alhaji Ibrahim, Sa’idu Isah Abubaka, <a href="http://dx.doi.org/10.4236/apm.2016.66028">Aunu Integer Sequence as Non-Associative Structure and Their Graph Theoretic Properties</a>, Advances in Pure Mathematics, 2016, 6, 409-419.
%H Ju. V. Linnik. <a href="http://mi.mathnet.ru/izv3674">An asymptotic formula in an additive problem of Hardy and Littlewood</a> (Russian). Izv. Akad. Nauk SSSR, ser. math., 24:629-706, 1960. Cited in Matomäki 2007.
%H Kaisa Matomäki, <a href="http://users.utu.fi/ksmato/papers/Primesm2n2p1.pdf">Prime numbers of the form p = m^2+n^2+1 in short intervals</a>, Acta Arith. 128 (2007), pp. 193-200.
%H Yu-Chen Sun and Hao Pan, <a href="https://arxiv.org/abs/1708.08629">The Green-Tao theorem for primes of the form x^2+y^2+1</a>, arXiv preprint arXiv:1708.08629 [math.NT], 2017.
%t iMax=7!; a=Floor[Sqrt[iMax]]; lst={}; Do[Do[p=x^2+y^2+1; If[PrimeQ@p&&p<=iMax,AppendTo[lst,p]],{y,1,a}],{x,1,a}]; Union[lst] (* _Vladimir Joseph Stephan Orlovsky_, Aug 11 2009 *)
%o (PARI) list(lim)=my(v=List(),t); lim\=1; for(x=1,sqrtint(lim-2), forstep(y=2-x%2,min(x,sqrtint(lim-x^2-1)), 2, if(isprime(t=x^2+y^2+1), listput(v,t)))); vecsort(Vec(v),,8) \\ _Charles R Greathouse IV_, Jun 13 2012
%Y Cf. A079545.
%K nonn
%O 1,1
%A _N. J. A. Sloane_, Jan 23 2003
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