A239451 a(n) = |{1 < k < sqrt(n)*log(n): prime(n) + C(prime(k)-1, (prime(k)-1)/2) is prime}|, where C(2j,j) = (2j)!/(j!)^2.

0, 0, 0, 0, 2, 1, 4, 1, 3, 4, 2, 2, 6, 1, 5, 6, 6, 3, 4, 2, 3, 2, 4, 6, 5, 6, 3, 5, 4, 3, 4, 7, 8, 4, 3, 4, 6, 4, 6, 7, 8, 4, 10, 5, 6, 3, 2, 3, 7, 4, 5, 8, 3, 7, 9, 5, 8, 3, 9, 5, 2, 4, 5, 6, 5, 6, 5, 3, 11, 7, 6, 7, 6, 4, 6, 6, 7, 6, 4, 4

Offset: 1

Author

Zhi-Wei Sun, Mar 19 2014

Keywords

• nonn

Comments

Conjecture: (i) a(n) > 0 for all n > 4.

(ii) For any integer n > 4, there is an odd prime p < prime(n) with n*C(p-1,(p-1)/2) + 1 prime.

(iii) For every n = 2, 3, ..., there is a positive integer k < 5*sqrt(n)/3 with n*C(2*k,k) - 1 prime.

(iv) For any integer n > 1, k!*n - 1 (or k!*n + 1) is prime for some k = 1, ..., n.

We have verified that a(n) > 0 for all n = 5, ..., 10^7.

Examples

			a(6) = 1 since prime(6) + C(prime(3)-1, (prime(3)-1)/2) = 13 + C(4, 2) = 13 + 6 = 19 is prime.
a(8) = 1 since prime(8) + C(prime(5)-1, (prime(5)-1)/2) = 19 + C(10, 5) = 19 + 252 = 271 is prime.
a(14) = 1 since prime(14) + C(prime(6)-1, (prime(6)-1)/2) = 43 + C(12, 6) = 43 + 924 = 967 is prime.
a(7597) = 1 since prime(7597) + C(prime(686)-1, (prime(686)-1)/2) = 77323 + C(5146, 2573) is prime.
a(193407) = 2 since prime(193407) + C(prime(3212)-1, (prime(3212)-1)/2) =  2652113 + C(29586, 14793) and prime(193407) + C(prime(5348)-1, (prime(5348)-1)/2) = 2652113 + C(52312, 26156) are both prime.
a(4517422) > 0 since prime(4517422) + C(prime(6918)-1, (prime(6918)-1)/2) = 77233291 + C(69778, 34889) is prime.
a(4876885) > 0 since prime(4876885) + C(prime(8904)-1, (prime(8904)-1)/2) = 83778493 + C(92202, 46101) is prime.
a(5887242) > 0 since prime(5887242) + C(prime(5678)-1, (prime(5678)-1)/2) = 102316597 + C(55930, 27965) is prime.
a(8000871) > 0 since prime(8000871) + C(prime(4797)-1, (prime(4797)-1)/2) = 141667111 + C(46410, 23205) is prime.
		

Links

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

Crossrefs

Cf. A000040, A000142, A000984, A231516, A233183, A233206.

Programs

Mathematica

p[n_,k_]:=PrimeQ[Prime[n]+Binomial[Prime[k]-1,(Prime[k]-1)/2]]
a[n_]:=Sum[If[p[n,k],1,0],{k,2,Ceiling[Sqrt[n]*Log[n]]-1}]
Table[a[n],{n,1,80}]