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Number of odd primes p < 2*n with prime(n*(p-1)/2)^2 - 2 prime.
5

%I #9 Mar 01 2014 09:52:53

%S 0,1,2,3,2,3,1,4,3,2,2,4,2,2,2,3,2,2,4,5,2,2,1,8,2,2,3,3,2,2,4,4,5,6,

%T 2,5,4,3,3,7,2,2,8,8,5,4,6,3,3,7,6,5,3,3,9,4,8,3,5,3,1,5,6,4,6,7,7,8,

%U 6,6,2,7,1,5,9,7,5,6,5,7

%N Number of odd primes p < 2*n with prime(n*(p-1)/2)^2 - 2 prime.

%C Conjecture: (i) a(n) > 0 for all n > 1, and a(n) = 1 only for n = 2, 7, 23, 61, 73.

%C (ii) For any integer n > 1, there is an odd prime p < 2*n with prime(n*(p+1)/2)^2 - 2 prime.

%C Clearly, either part of the conjecture implies that there are infinitely many primes of the form p^2 - 2 with p prime.

%H Zhi-Wei Sun, <a href="/A238576/b238576.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(2) = 1 since 2 and prime(2*(3-1)/2)^2 - 2 = 3^2 - 2 = 7 are both prime.

%e a(7) = 1 since 5 and prime(7*(5-1)/2)^2 - 2 = 43^2 - 2 = 1847 are both prime.

%e a(23) = 1 since 29 and prime(23*(29-1)/2)^2 - 2 = 2137^2 - 2 = 4566767 are both prime.

%e a(61) = 1 since 43 and prime(61*(43-1)/2)^2 - 2 = 10463^2 - 2 = 109474367 are both prime.

%e a(73) = 1 since 7 and prime(73*(7-1)/2)^2 - 2 = 1367^2 - 2 = 1868687 are both prime.

%t p[k_,n_]:=PrimeQ[Prime[(Prime[k]-1)/2*n]^2-2]

%t a[n_]:=Sum[If[p[k,n],1,0],{k,2,PrimePi[2n-1]}]

%t Table[a[n],{n,1,80}]

%Y Cf. A000040, A049002, A062326, A237413, A237578, A238573.

%K nonn

%O 1,3

%A _Zhi-Wei Sun_, Mar 01 2014