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%I #18 Apr 10 2013 19:18:26
%S 31,59,79,179,191,229,251,311,389,401,479,491,541,569,719,809,971,
%T 1019,1061,1109,1151,1249,1301,1409,1451,1499,1619,1931,1949,2111,
%U 2141,2339,2591,2609,2711,2801,2939,3089,3371,3389,3449,3881,4021,4091,4211,4391,4451,4679,5039,5051
%N Primes of the form p^2 + 3pq + q^2 with p and q prime.
%C It is easy to see that a(n) is congruent to 1 or 9 modulo 10. For each n there is a unique pair of primes p < q such that p^2 + 3pq + q^2 = a(n).
%C This sequence is of particular interest due to Zhi-Wei Sun's surprising conjecture related to A218754. That conjecture implies that this sequence is infinite.
%H Zhi-Wei Sun, <a href="/A218771/b218771.txt">Table of n, a(n) for n = 1..10000</a>
%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1211.1588">Conjectures involving primes and quadratic forms</a>, arXiv preprint arXiv:1211.1588, 2012.
%e a(1)=31 since 2^2 + 3*2*3 + 3^2 = 31 and 2,3,31 are prime.
%t SQ[n_]:=IntegerQ[Sqrt[n]]
%t i=0; Do[Do[If[SQ[4Prime[n]+5Prime[k]^2] && PrimeQ[(Sqrt[4Prime[n] + 5Prime[k]^2] - 3Prime[k])/2] == True, i=i+1; Print[i," ", Prime[n]]; Goto[aa]], {k,1,PrimePi[Sqrt[Prime[n]/5]]}];
%t Label[aa];Continue,{n,1,1000000}]
%o (PARI) list(lim)=my(v=List(),t);forprime(p=2, sqrtint(lim\4), forprime(q=p+1,sqrt(lim-p^2), if(isprime(t=p^2+3*p*q+q^2), listput(v,t), if(t>lim,break)))); vecsort(Vec(v)) \\ _Charles R Greathouse IV_, Nov 05 2012
%o (PARI) is_A218771(n,v=0)={ my(r,c=0); isprime(n) & forprime( q=1,sqrtint(n\5), issquare(4*n+5*q^2, &r) || next; isprime((r-3*q)/2) || next; v || return(1); v>1 & print1([q,(r-3*q)/2]","); c++);c} \\ - _M. F. Hasler_, Nov 05 2012
%Y Cf. A038872, A218754, A218585, A218654, A218656.
%K nonn
%O 1,1
%A _Zhi-Wei Sun_, Nov 05 2012
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