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a(n) = |{1 < k < sqrt(n)*log(n): prime(n) + C(prime(k)-1, (prime(k)-1)/2) is prime}|, where C(2j,j) = (2j)!/(j!)^2.
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%I #27 Mar 24 2014 09:05:49

%S 0,0,0,0,2,1,4,1,3,4,2,2,6,1,5,6,6,3,4,2,3,2,4,6,5,6,3,5,4,3,4,7,8,4,

%T 3,4,6,4,6,7,8,4,10,5,6,3,2,3,7,4,5,8,3,7,9,5,8,3,9,5,2,4,5,6,5,6,5,3,

%U 11,7,6,7,6,4,6,6,7,6,4,4

%N a(n) = |{1 < k < sqrt(n)*log(n): prime(n) + C(prime(k)-1, (prime(k)-1)/2) is prime}|, where C(2j,j) = (2j)!/(j!)^2.

%C Conjecture: (i) a(n) > 0 for all n > 4.

%C (ii) For any integer n > 4, there is an odd prime p < prime(n) with n*C(p-1,(p-1)/2) + 1 prime.

%C (iii) For every n = 2, 3, ..., there is a positive integer k < 5*sqrt(n)/3 with n*C(2*k,k) - 1 prime.

%C (iv) For any integer n > 1, k!*n - 1 (or k!*n + 1) is prime for some k = 1, ..., n.

%C We have verified that a(n) > 0 for all n = 5, ..., 10^7.

%H Zhi-Wei Sun, <a href="/A239451/b239451.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(6) = 1 since prime(6) + C(prime(3)-1, (prime(3)-1)/2) = 13 + C(4, 2) = 13 + 6 = 19 is prime.

%e a(8) = 1 since prime(8) + C(prime(5)-1, (prime(5)-1)/2) = 19 + C(10, 5) = 19 + 252 = 271 is prime.

%e a(14) = 1 since prime(14) + C(prime(6)-1, (prime(6)-1)/2) = 43 + C(12, 6) = 43 + 924 = 967 is prime.

%e a(7597) = 1 since prime(7597) + C(prime(686)-1, (prime(686)-1)/2) = 77323 + C(5146, 2573) is prime.

%e a(193407) = 2 since prime(193407) + C(prime(3212)-1, (prime(3212)-1)/2) = 2652113 + C(29586, 14793) and prime(193407) + C(prime(5348)-1, (prime(5348)-1)/2) = 2652113 + C(52312, 26156) are both prime.

%e a(4517422) > 0 since prime(4517422) + C(prime(6918)-1, (prime(6918)-1)/2) = 77233291 + C(69778, 34889) is prime.

%e a(4876885) > 0 since prime(4876885) + C(prime(8904)-1, (prime(8904)-1)/2) = 83778493 + C(92202, 46101) is prime.

%e a(5887242) > 0 since prime(5887242) + C(prime(5678)-1, (prime(5678)-1)/2) = 102316597 + C(55930, 27965) is prime.

%e a(8000871) > 0 since prime(8000871) + C(prime(4797)-1, (prime(4797)-1)/2) = 141667111 + C(46410, 23205) is prime.

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

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

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

%Y Cf. A000040, A000142, A000984, A231516, A233183, A233206.

%K nonn

%O 1,5

%A _Zhi-Wei Sun_, Mar 19 2014