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%I #7 Mar 01 2014 14:11:54
%S 0,1,2,1,1,2,2,4,1,1,5,3,1,3,1,2,4,3,3,2,2,3,4,3,1,5,3,1,3,2,4,5,2,2,
%T 2,3,3,6,3,3,4,2,4,5,3,4,5,3,2,6,2,3,8,1,1,5,5,3,5,4,4,6,2,3,3,4,3,7,
%U 3,1,7,4,4,5,4,3,8,4,1,7
%N Number of primes p < 2*n with 2*pi(p) + 1 and p*(2n-1) - 2 both prime, where pi(.) is given by A000720.
%C Conjecture: (i) a(n) > 0 for all n > 1, and a(n) = 1 for no n > 195.
%C (ii) For any integer n > 1, there is a prime p < 2*n with 2*pi(p) + 1 (or 2*pi(p) - 1) and 2*n + p both prime.
%C Part (i) of this conjecture is an extension of the conjecture in A238580.
%H Zhi-Wei Sun, <a href="/A238597/b238597.txt">Table of n, a(n) for n = 1..10000</a>
%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1402.6641">Problems on combinatorial properties of primes</a>, arXiv:1402.6641, 2014.
%e a(9) = 1 since 5, 2*pi(5) + 1 = 2*3 + 1 = 7 and 5*(2*9-1) - 2 = 5*17 - 2 = 83 are all prime.
%e a(28) = 1 since 3, 2*pi(3) + 1 = 2*2 + 1 = 5 and 3*(2*28-1) - 2 = 3*55 - 2 = 163 are all prime.
%e a(195) = 1 since 71, 2*pi(71) + 1 = 2*20 + 1 = 41 and 71*(2*195-1) - 2 = 27617 are all prime.
%t p[n_,k_]:=p[n,k]=PrimeQ[k]&&PrimeQ[2*PrimePi[k]+1]&&PrimeQ[k*(2n-1)-2]
%t a[n_]:=a[n]=Sum[If[p[n,k],1,0],{k,1,2n-1}]
%t Table[a[n],{n,1,80}]
%Y Cf. A000040, A000720, A237284, A238580.
%K nonn
%O 1,3
%A _Zhi-Wei Sun_, Mar 01 2014
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