A182663 Number of ordered ways to write n = p + q with q > 0 such that p and 2*p + prime(q) are both prime.

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

Offset: 1

Author

Zhi-Wei Sun, Feb 01 2014

Keywords

• nonn

Comments

Conjecture: (i) a(n) > 0 for all n > 5.

(ii) Any integer n > 5 can be written as p + q (q > 0) with p and prime(p) + 2*q both prime.

(iii) Each integer n > 10 can be written as p + q (q > 0) with p and prime(p)^2 + 4*q^2 both prime.

Examples

			 a(11) = 1 since 11 = 5 + 6 with 5 and 2*5 + prime(6) = 10 + 13 = 23 both prime.
a(98) = 1 since 98 = 19 + 79 with 19 and 2*19 + prime(79) = 38 + 401 = 439 both prime.
		

Links

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

Crossrefs

Cf. A000040, A182662, A236831.

Programs

Mathematica

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