A233183 Number of ways to write n = k + m with 0 < k < m such that C(2*k, k) + prime(m) is prime.

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, 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, 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

Offset: 1

Author

Zhi-Wei Sun, Dec 05 2013

Keywords

• nonn

Comments

Conjecture: a(n) > 0 for all n > 2.

We have verified this for n up to 10^8.

Examples

			a(6) = 2 since 6 = 1 + 5 = 2 + 4 with C(2*1, 1) + prime(5) = C(2*2, 2) + prime(4) = 13 prime.
a(9) = 1 since 9 = 2 + 7 with C(2*2, 2) + prime(7) = 6 + 17 = 23 prime.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..3000
• Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014

Crossrefs

Cf. A000040, A000984, A231201, A233150.

Programs

Mathematica

a[n_]:=Sum[If[PrimeQ[Binomial[2k,k]+Prime[n-k]],1,0],{k,1,(n-1)/2}]
Table[a[n],{n,1,100}]