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%I #17 Aug 16 2015 12:07:21
%S 31,191,541,809,1153,1301,2221,3037,3847,4049,4159,5441,8243,10177,
%T 12277,13681,14783,15619,17903,19463,20897,22697,24517,25163,25847,
%U 25849,26633,26647,27329,27407,28051,32653,35059,35747,36341,36527,37369,37811,38609,40949,42737,46679,51061,51607,54443,54679,56113,57637,60887,61493
%N Primes p such that prime(p)^2 - 2 = prime(q) for some prime q.
%C Conjecture: The sequence has infinitely many terms. In general, for any integers a,b,c with a>0 and gcd(a,b,c)=1, if b^2-4*a*c is not a square, a+b+c is odd, and gcd(b,a+c) is not divisible by 3, then there are infinitely many prime pairs {p,q} such that a*prime(p)^2+b*prime(p)+c = prime(q).
%D Zhi-Wei Sun, Problems on combinatorial properties of primes, in: M. Kaneko, S. Kanemitsu and J. Liu (eds.), Number Theory: Plowing and Starring through High Wave Forms, Proc. 7th China-Japan Seminar (Fukuoka, Oct. 28 - Nov. 1, 2013), Ser. Number Theory Appl., Vol. 11, World Sci., Singapore, 2015, pp. 169-187.
%H Zhi-Wei Sun, <a href="/A261354/b261354.txt">Table of n, a(n) for n = 1..100</a>
%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1402.6641">Problems on combinatorial properties of primes</a>, arXiv:1402.6641 [math.NT], 2014.
%e a(1) = 31 since 31 is a prime, and prime(31)^2-2 = 127^2-2 = 16127 = prime(1877) with 1877 prime.
%t PQ[n_]:=PrimeQ[n]&&PrimeQ[PrimePi[n]]
%t f[k_]:=Prime[Prime[k]]^2-2
%t n=0;Do[If[PQ[f[k]],n=n+1;Print[n," ",Prime[k]]],{k,1,6200}]
%Y Cf. A000040, A049002, A062326, A237413, A260120, A261281, A261352, A261361.
%K nonn
%O 1,1
%A _Zhi-Wei Sun_, Aug 15 2015
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