A237715 Number of ordered ways to write n = p + q (q > 0) with p, prime(p) - p + 1 and prime(prime(q)) - prime(q) + 1 all prime.

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

Offset: 1

Author

Zhi-Wei Sun, Mar 06 2014

Keywords

• nonn

Comments

Conjecture: a(n) > 0 for all n > 2, and a(n) = 1 only for n = 3.

Examples

			a(3) = 1 since 3 = 2 + 1 with 2, prime(2) - 2 + 1 = 3 - 1 = 2 and prime(prime(1)) - prime(1) + 1 = prime(2) - 2 + 1  = 2 all prime.
a(7) = 2 since 7 = 3 + 4 with 3, prime(3) - 3 + 1 = 5 - 2 = 3 and prime(prime(4)) - prime(4) + 1 = prime(7) - 7 + 1 = 17 - 6 = 11 are all prime, and 7 = 5 + 2 with 5, prime(5) - 5 + 1 = 11 - 4 = 7 and prime(prime(2)) - prime(2) + 1 = prime(3) - 3 + 1 = 5 - 2 = 3 all prime.
		

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, A234694, A234695, A238766, A238776, A238814.

Programs

Mathematica

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