A236831 Number of ordered ways to write n = p + q with q > 0 such that p, p + 2 and p + prime(q) + 1 are all prime.

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

Offset: 1

Author

Zhi-Wei Sun, Jan 31 2014

Keywords

• nonn

Comments

Conjecture: a(n) > 0 for all n > 13.

Examples

			a(12) = 1 since 12 = 5 + 7 with 5, 5 + 2 = 7 and 5 + prime(7) + 1 = 5 + 17 + 1 = 23 all prime.
a(85) = 1 since 85 = 29 + 56 with 29, 29 + 2 = 31 and 29 + prime(56) + 1 = 29 + 263 + 1 = 293 all prime.
		

Links

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

Crossrefs

Cf. A000040, A001359, A006512, A236531.

Programs

Mathematica

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