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%I #12 Jan 27 2014 02:53:33
%S 3,1949,4217,8219,9929,22091,23537,28097,38711,41609,50051,60899,
%T 68111,72227,74159,79631,115151,122399,127679,150959,155537,266687,
%U 267611,270551,271499,284741,306347,428297,433661,444287
%N Primes p with p + 2, prime(p) + 2 and prime(prime(p)) + 2 all prime.
%C Conjecture: For any positive integer m, there are infinitely many chains p(1) < p(2) < ... < p(m) of m primes with p(k) + 2 prime for all k = 1,...,m such that p(k + 1) = prime(p(k)) for every 0 < k < m.
%H Zhi-Wei Sun, <a href="/A236481/b236481.txt">Table of n, a(n) for n = 1..5000</a>
%H Zhi-Wei Sun, <a href="http://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;e10c069b.1401">A new kind of prime chains</a>, a message to Number Theory List, Jan. 20, 2014.
%e a(1) = 3 since 3, 3 + 2 = 5, prime(3) + 2 = 7 and prime(prime(3)) + 2 = prime(5) + 2 = 13 are all prime, but 2 + 2 = 4 is composite.
%t p[n_]:=p[n]=PrimeQ[n+2]&&PrimeQ[Prime[n]+2]&&PrimeQ[Prime[Prime[n]]+2]
%t n=0;Do[If[p[Prime[m]],n=n+1;Print[n," ",Prime[m]]],{m,1,10^6}]
%Y Cf. A000010, A000040, A001359, A006512, A235925, A236066, A236457, A236458, A236468, A236470, A236480, A236482, A236484.
%K nonn
%O 1,1
%A _Zhi-Wei Sun_, Jan 26 2014
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