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

1,4

Comments

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

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.

Example

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.

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}]

Crossrefs

Cf. A000040, A234694, A234695, A238766, A238776, A238814.

Sequence in context: A101037 A002199 A218829 * A238458 A182744 A308355

Adjacent sequences: A237712 A237713 A237714 * A237716 A237717 A237718

Keyword

nonn

Author

Zhi-Wei Sun, Mar 06 2014

Status

approved