A233206 - OEIS

%I #15 Apr 04 2014 18:48:32

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

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

%U 5,9,6,2,6,7,3,8,8,9,9,7,6,10,8,8,11,7,7,4,6,8,8,5,8,5,8,14,8,7,10,8

%N Number of ways to write n = k + m (0 < k <= m) with k! + prime(m) prime.

%C Conjecture: a(n) > 0 for all n > 3.

%C We have verified this for n up to 10^7. For n = 1356199, the least positive integer k with k! + prime(n-k) prime is 4496. For n = 7212995, the smallest positive integer k with k! + prime(n-k) prime is 4507.

%H Zhi-Wei Sun, <a href="/A233206/b233206.txt">Table of n, a(n) for n = 1..2000</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(6) = 1 since 6 = 3 + 3 with 3! + prime(3) = 6 + 5 = 11 prime.

%e a(11) = 1 since 11 = 4 + 7 with 4! + prime(7) = 24 + 17 = 41 prime.

%t a[n_]:=Sum[If[PrimeQ[k!+Prime[n-k]],1,0],{k,1,n/2}]

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

%Y Cf. A000040, A000142, A231201, A231516, A231557, A231561, A231631, A233150, A233183, A233204.

%K nonn

%O 1,7

%A _Zhi-Wei Sun_, Dec 05 2013