A233204 - OEIS

%I #10 Apr 04 2014 18:47:52

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

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

%U 7,7,6,6,2,7,8,7,7,5,11,3,4,8,2,7,8,6,9,7,6,10,11,4,5,8,4,8,8,6,7,9

%N Number of ways to write n = k + m with 0 < k < m such that 2^k * prime(m) + 3 is prime.

%C Conjecture: (i) a(n) > 0 for all n > 3. Also, any integer n > 2 can be written as k + m (0 < k <= m) with 2^k*prime(m) - 3 prime.

%C (ii) Any integer n > 6 can be written as k + m (0 < k < m) with prime(k) + 6 and prime(m) + 6 both prime. Each integer n > 4 can be written as k + m (0 < k < m) with prime(k) + 2 and prime(m) + 6 both prime.  Also, for every integer n > 3 not among 11, 21, 32, 49, 171, there is a positive integer k < n with prime(k) + 2 and prime(n-k) + 2 both prime.

%H Zhi-Wei Sun, <a href="/A233204/b233204.txt">Table of n, a(n) for n = 1..8000</a>

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

%p a(6) = 1 since 6 = 2 + 4 with 2^2*prime(4) + 3 = 4*7 + 3 = 31 prime.

%p a(22) = 1 since 22 = 1 + 21 with 2^1*prime(21) + 3 = 2*73 + 3 = 149 prime.

%t a[n_]:=Sum[If[PrimeQ[2^k*Prime[n-k]+3],1,0],{k,1,(n-1)/2}]

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

%Y Cf. A000040, A000079, A231201, A231561, A233150, A233183.

%K nonn

%O 1,5

%A _Zhi-Wei Sun_, Dec 05 2013