A236566 Number of ordered ways to write 2*n = p + q with p, q and prime(p + 2) + 2 all prime.

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

Offset: 1

Author

Zhi-Wei Sun, Jan 28 2014

Keywords

• nonn

Comments

Conjecture: (i) a(n) > 0 for all n > 2.

(ii) If n > 30, then 2*n + 1 can be written as 2*p + q with p, q and prime(p + 2) + 2 all prime.

Part (i) implies both the Goldbach conjecture and the twin prime conjecture. If all primes p with prime(p + 2) + 2 are smaller than an even number N > 2, then for any such a prime p the number N! + N - p is in the interval (N!, N! + N) and hence not prime.

Similarly, part (ii) implies both Lemoine's conjecture (cf. A046927) and the twin prime conjecture.

We have verified part (i) of the conjecture for n up to 2*10^8.

Examples

			a(10) = 1 since 2*10 = 3 + 17 with 3, 17 and prime(3 + 2) + 2 = 11 + 2 = 13 all prime.
a(589) = 1 since 2*589 = 577 + 601 with 577, 601 and prime(577 + 2) + 2 = 4229 + 2 = 4231 all prime.
		

Links

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

Crossrefs

Cf. A000040, A001359, A002372, A002375, A006512, A046927, A236531.

Programs

Mathematica

p[m_]:=PrimeQ[Prime[m+2]+2]
a[n_]:=Sum[If[p[Prime[k]]&&PrimeQ[2n-Prime[k]],1,0],{k,1,PrimePi[2n-1]}]
Table[a[n],{n,1,100}]