A236241 a(n) = |{0 < k < n: m = phi(k) + phi(n-k)/8 is an integer with C(2*m, m) + prime(m) prime}|, where C(2*m, m) = (2*m)!/(m!)^2, and phi(.) is Euler's totient function.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 3, 4, 2, 3, 4, 5, 2, 2, 2, 3, 4, 3, 2, 4, 4, 6, 3, 5, 8, 9, 6, 6, 4, 5, 5, 4, 5, 6, 6, 4, 4, 4, 10, 9, 7, 4, 4, 5, 7, 2, 2, 3, 7, 7, 5, 7, 6, 7, 5, 4, 7, 5, 5, 3, 8, 6, 4, 6, 5, 8, 9, 5, 4, 3

Offset: 1

Author

Zhi-Wei Sun, Jan 20 2014

Keywords

• nonn

Comments

Conjecture: a(n) > 0 for every n = 20, 21, ... .

We have verified this for n up to 75000.

The conjecture implies that there are infinitely many primes of the form C(2*m, m) + prime(m).

See A236245 for primes of the form C(2*m, m) + prime(m). See also A236242 for a list of known numbers m with C(2*m, m) + prime(m) prime.

Examples

			a(20) = 1 since phi(5) + phi(15)/8 = 4 + 1 = 5 with C(2*5,5) + prime(5) = 252 + 11 = 263 prime.
a(330) = 1 since phi(211) + phi(330-211)/8 = 210 + 96/8 = 222 with C(2*222,222) + prime(222) = C(444,222) + 1399 prime.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
• Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014

Crossrefs

Cf. A000010, A000040, A000984, A236242, A236245, A236248, A236249, A236256.

Programs

Mathematica

p[n_]:=IntegerQ[n]&&PrimeQ[Binomial[2n,n]+Prime[n]]
f[n_,k_]:=EulerPhi[k]+EulerPhi[n-k]/8
a[n_]:=Sum[If[p[f[n,k]],1,0],{k,1,n-1}]
Table[a[n],{n,1,100}]