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%I #6 Apr 09 2014 04:28:18
%S 10909,67247,185869,408379,511111,1297061,1730461,1732333,2135347,
%T 2266079,2316203,2978917,3477737,4337257,4495739,4691849,6108461,
%U 6407971,6591163,7462589,7909507,8165039,8298337,8948509,11144083,11961373,15019049,16074059,16732561,19316263
%N Primes p with pi(p), pi(2*p), pi(3*p) and pi(4*p) all prime, where pi(x) denotes the number of primes not exceeding x.
%C Conjecture: For any positive integer n, there are infinitely many primes p with pi(k*p) (k = 1,...,n) all prime.
%H Zhi-Wei Sun, <a href="/A240604/b240604.txt">Table of n, a(n) for n = 1..2280</a>
%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1402.6641">Problems on combinatorial properties of primes</a>, preprint, arXiv:1402.6641, 2014.
%e a(1) = 10909 with 10909, pi(10909) = 1327, pi(2*10909) = 2447, pi(3*10909) = 3511 and pi(4*10909) = 4547 all prime.
%t p[j_,k_]:=p[j,k]=PrimeQ[PrimePi[j*Prime[Prime[k]]]]
%t p[k_]:=p[k]=p[2,k]&&p[3,k]&&p[4,k]
%t m=0;Do[If[p[k],m=m+1;Print[m," ",Prime[Prime[k]]]],{k,1,95041}]
%Y Cf. A000040, A000720, A006450, A237578, A239428, A239430, A239617.
%K nonn
%O 1,1
%A _Zhi-Wei Sun_, Apr 09 2014
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