A236242 Numbers m with C(2*m, m) + prime(m) prime, where C(2*m, m) = (2*m)!/(m!)^2.

5, 6, 7, 8, 12, 13, 19, 69, 91, 102, 116, 119, 171, 198, 216, 222, 278, 299, 338, 584, 722, 774, 874, 978, 1004, 1163, 1268, 1492, 1836, 1932, 1966, 2982, 3508, 3964, 4264, 4894, 5028, 8236, 8552, 8639, 12749, 14017, 14402, 18150, 18321, 18514, 18979, 20935, 21815, 21828, 21890, 30734

Offset: 1

Author

Zhi-Wei Sun, Jan 20 2014

Keywords

• nonn

Comments

According to the conjecture in A236241, this sequence should have infinitely many terms. The prime C(2*a(52),a(52)) + prime(a(52)) = C(61468, 30734) + prime(30734) has 18502 decimal digits.

For primes of the form C(2*m, m) + prime(m), see A236245.

See also A236248 for a similar sequence.

Examples

			a(1) = 5 since C(2*1,1) + prime(1) = 4, C(2*2,2) + prime(2) = 9, C(2*3,3) + prime(3) = 25 and C(2*4,4) + prime(4) = 77 are all composite, but C(2*5,5) + prime(5) = 252 + 11 = 263 is prime.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..52
• Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014-2016.

Crossrefs

Cf. A000040, A000984, A236241, A236245, A236248, A236249, A236256.

Programs

Mathematica

n=0;Do[If[PrimeQ[Binomial[2m,m]+Prime[m]],n=n+1;Print[n," ",m]],{m,1,10000}]
Select[Range[9000],PrimeQ[Binomial[2#,#]+Prime[#]]&] (* Harvey P. Dale, Jan 18 2016 *)

Extensions

a(41)-a(52) from bfile by Robert Price, Aug 31 2019