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

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, 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, 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

Offset: 1

Author

Zhi-Wei Sun, Dec 11 2013

Keywords

• nonn

Comments

Conjectures:

(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.

(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.

Examples

			a(8) = 1 since prime(4)*prime(4) - 6 = 7*7 - 6 = 43 is prime.
a(10) = 1 since prime(3)*prime(7) - 6 = 5*17 - 6 = 79 is prime.
a(16) = 1 since prime(3)*prime(13) - 6 = 5*41 - 6 = 199 is prime.
a(20) = 1 since prime(7)*prime(13) - 6 = 17*41 - 6 = 691 is prime.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
• Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014

Crossrefs

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

Programs

Mathematica

PQ[n_]:=n>0&&PrimeQ[n]
a[n_]:=Sum[If[PQ[Prime[k]*Prime[n-k]-6],1,0],{k,1,n/2}]
Table[a[n],{n,1,100}]