A238580 a(n) = |{0 < k <= n: 2*k + 1 and prime(k)*prime(n) - 2 are both prime}|.

1, 1, 2, 1, 2, 2, 1, 1, 3, 1, 2, 3, 2, 3, 3, 3, 2, 4, 2, 3, 3, 3, 2, 3, 2, 2, 3, 1, 1, 5, 3, 4, 3, 1, 4, 3, 1, 5, 4, 4, 2, 4, 5, 4, 5, 2, 5, 5, 3, 2, 4, 2, 4, 5, 3, 5, 2, 7, 4, 5, 2, 5, 4, 8, 4, 6, 5, 6, 5, 2, 5, 4, 3, 6, 2, 5, 1, 5, 8, 4

Offset: 1

Author

Zhi-Wei Sun, Mar 01 2014

Keywords

• nonn

Comments

Conjecture: a(n) > 0 for all n > 0, and a(n) = 1 only for n = 1, 2, 4, 7, 8, 10, 28, 34,37, 77.

Note that a prime p with p + 2 a product of at most two primes is called a Chen prime.

Examples

			a(7) = 1 since 2*3 + 1 = 7 and prime(3)*prime(7) - 2 = 5*17 - 2 = 83 are both prime.
a(8) = 1 since 2*8 + 1 = 17 and prime(8)*prime(8) - 2 = 19^2 - 2 = 359 are both prime.
a(77) = 1 since 2*20 + 1 = 41 and prime(20)*prime(77) - 2 = 71*389 - 2 = 27617 are both prime.
		

References

• J.-R. Chen, On the representation of a large even integer as the sum of a prime and a product of at most two primes, Sci. Sinica 16(1973), 157-176.

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, A109611, A233529.

Programs

Mathematica

p[n_,k_]:=PrimeQ[2k+1]&&PrimeQ[Prime[n]*Prime[k]-2]
a[n_]:=Sum[If[p[n,k],1,0],{k,1,n}]
Table[a[n],{n,1,80}]