A233529 - OEIS

%I #8 Apr 05 2014 22:28:02

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

%T 6,6,4,8,6,7,7,8,7,5,5,5,7,8,6,13,9,5,3,9,6,8,11,5,9,9,10,8,9,14,9,10,

%U 13,11,6,9,12,10,12,14,10,12,7,13,9,7,7,15,12,6,10,11,12,12,9,18,15,14,11,10,10,8,13,21,9,14

%N a(n) = |{0 < k <= n/2: prime(k)*prime(n-k) - 6 is prime}|.

%C Conjectures:

%C (i) a(n) > 0 for all n > 5. Also, for any n > 5, 2*prime(k)*prime(n-k) - 3 is prime for some 0 < k < n.

%C (ii) For any n > 1 not among 3, 9, 13, 26, there is a positive integer k < n with prime(k)*prime(n-k) - 2 prime.  For any n > 2 not among 8, 23, 33, there is a positive integer k < n with prime(k)*prime(n-k) - 4 prime.

%H Zhi-Wei Sun, <a href="/A233529/b233529.txt">Table of n, a(n) for n = 1..10000</a>

%H Z.-W. Sun, <a href="http://arxiv.org/abs/1402.6641">Problems on combinatorial properties of primes</a>, arXiv:1402.6641, 2014

%e a(8) = 1 since prime(4)*prime(4) - 6 = 7*7 - 6 = 43 is prime.

%e a(10) = 1 since prime(3)*prime(7) - 6 = 5*17 - 6 = 79 is prime.

%e a(16) = 1 since prime(3)*prime(13) - 6 = 5*41 - 6 = 199 is prime.

%e a(20) = 1 since prime(7)*prime(13) - 6 = 17*41 - 6 = 691 is prime.

%t PQ[n_]:=n>0&&PrimeQ[n]

%t a[n_]:=Sum[If[PQ[Prime[k]*Prime[n-k]-6],1,0],{k,1,n/2}]

%t Table[a[n],{n,1,100}]

%Y Cf. A000040, A232465, A232502, A232861, A233150, A233204, A233206, A233439.

%K nonn

%O 1,9

%A _Zhi-Wei Sun_, Dec 11 2013