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%I #8 Apr 04 2014 18:47:25
%S 0,0,1,1,1,2,1,3,1,2,2,3,3,2,4,4,3,7,3,4,4,4,5,2,3,5,5,3,7,7,6,2,5,3,
%T 7,6,9,6,5,5,6,8,6,6,2,12,6,7,6,9,4,5,7,5,3,7,8,8,6,5,7,9,10,4,9,6,7,
%U 7,8,6,10,8,6,6,8,5,5,10,8,10,5,9,8,15,8,12,3,12,9,10,9,10,5,11,12,8,3,12,12,8
%N Number of ways to write n = k + m with 0 < k < m such that C(2*k, k) + prime(m) is prime.
%C Conjecture: a(n) > 0 for all n > 2.
%C We have verified this for n up to 10^8.
%H Zhi-Wei Sun, <a href="/A233183/b233183.txt">Table of n, a(n) for n = 1..3000</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) = 2 since 6 = 1 + 5 = 2 + 4 with C(2*1, 1) + prime(5) = C(2*2, 2) + prime(4) = 13 prime.
%e a(9) = 1 since 9 = 2 + 7 with C(2*2, 2) + prime(7) = 6 + 17 = 23 prime.
%t a[n_]:=Sum[If[PrimeQ[Binomial[2k,k]+Prime[n-k]],1,0],{k,1,(n-1)/2}]
%t Table[a[n],{n,1,100}]
%Y Cf. A000040, A000984, A231201, A233150.
%K nonn
%O 1,6
%A _Zhi-Wei Sun_, Dec 05 2013
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