A236256 a(n) = |{0 < k < n: m = phi(k) + phi(n-k)/4 is an integer with C(2*m, m) - prime(m) prime}|, where C(2*m, m) = (2*m)!/(m!)^2.

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

Offset: 1

Author

Zhi-Wei Sun, Jan 21 2014

Keywords

• nonn

Comments

Conjecture: a(n) > 0 for all n > 410.

This implies that there are infinitely many positive integers m with C(2*m, m) - prime(m) prime. We have verified the conjecture for n up to 51000.

See A236248 for a list of known numbers m with C(2*m, m) - prime(m) prime.

See also A236249 for those primes of the form C(2*m, m) - prime(m).

Examples

			a(12) = 1 since phi(2) + phi(10)/4 = 1 + 1 = 2 with C(2*2, 2) - prime(2) = 6 - 3 = 3 prime.
a(33) = 1 since phi(1) + phi(32)/4 = 1 + 4 = 5 with C(2*5, 5) - prime(5) = 252 - 11 = 241 prime.
a(697) = 1 since phi(452) + phi(697-452)/4 = 224 + 42 = 266 with C(2*266, 266) - prime(266) = C(532, 266) - 1699 prime.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000010, A000040, A000984, A236241, A236242, A236245, A236248, A236249.

Programs

Mathematica

p[n_]:=IntegerQ[n]&&PrimeQ[Binomial[2n,n]-Prime[n]]
f[n_,k_]:=EulerPhi[k]+EulerPhi[n-k]/4
a[n_]:=Sum[If[p[f[n,k]],1,0],{k,1,n-1}]
Table[a[n],{n,1,100}]