A235187 Number of ordered ways to write 2*n = p + q with p, q and prime(p) + q - 1 all prime.

0, 0, 1, 1, 2, 1, 1, 2, 2, 3, 2, 3, 2, 1, 2, 2, 2, 3, 1, 2, 5, 5, 2, 5, 3, 2, 5, 2, 1, 6, 2, 4, 4, 1, 5, 3, 4, 3, 6, 6, 3, 5, 5, 2, 9, 3, 3, 7, 2, 4, 7, 6, 3, 6, 7, 5, 4, 4, 4, 12, 3, 2, 5, 3, 3, 9, 3, 1, 7, 4, 2, 8, 6, 3, 8, 3, 4, 7, 6, 3, 10, 3, 3, 10, 8, 3, 11, 5, 3, 10, 6, 1, 9, 8, 2, 7, 4, 3, 9, 4

Offset: 1

Author

Zhi-Wei Sun, Jan 04 2014

Keywords

• nonn

Comments

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

(ii) Any integer n > 4 can be written as p + q with q > 0 such that p and p - 1 + prime(q) are both prime.

(iii) Each integer n > 7 can be written as p + q with q > 0 such that prime(p) + sigma(q) is prime, where sigma(q) denotes the sum of all positive divisors of q.

Clearly, part (i) is stronger than Goldbach's conjecture.

Examples

			 a(7) = 1 since 2*7 = 7 + 7 with 7 and prime(7) + 7 - 1 = 17 + 6 = 23 both prime.
a(14) = 1 since 2*14 = 11 + 17 with 11, 17 and prime(11) + 16 = 47 all prime.
a(92) = 1 since 2*92 = 47 + 137 with 47, 137 and prime(47) + 136 = 347 all prime.
		

Links

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

Crossrefs

Cf. A000040, A000203, A045917, A234694.

Programs

Mathematica

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