cp's OEIS Frontend

A233529 a(n) = |{0 < k <= n/2: prime(k)*prime(n-k) - 6 is prime}|.

%I A233529 #8 Apr 05 2014 22:28:02
%S A233529 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 A233529 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 A233529 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 A233529 a(n) = |{0 < k <= n/2: prime(k)*prime(n-k) - 6 is prime}|.
%C A233529 Conjectures:
%C A233529 (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 A233529 (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 A233529 Zhi-Wei Sun, <a href="/A233529/b233529.txt">Table of n, a(n) for n = 1..10000</a>
%H A233529 Z.-W. Sun, <a href="http://arxiv.org/abs/1402.6641">Problems on combinatorial properties of primes</a>, arXiv:1402.6641, 2014
%e A233529 a(8) = 1 since prime(4)*prime(4) - 6 = 7*7 - 6 = 43 is prime.
%e A233529 a(10) = 1 since prime(3)*prime(7) - 6 = 5*17 - 6 = 79 is prime.
%e A233529 a(16) = 1 since prime(3)*prime(13) - 6 = 5*41 - 6 = 199 is prime.
%e A233529 a(20) = 1 since prime(7)*prime(13) - 6 = 17*41 - 6 = 691 is prime.
%t A233529 PQ[n_]:=n>0&&PrimeQ[n]
%t A233529 a[n_]:=Sum[If[PQ[Prime[k]*Prime[n-k]-6],1,0],{k,1,n/2}]
%t A233529 Table[a[n],{n,1,100}]
%Y A233529 Cf. A000040, A232465, A232502, A232861, A233150, A233204, A233206, A233439.
%K A233529 nonn
%O A233529 1,9
%A A233529 _Zhi-Wei Sun_, Dec 11 2013