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

1,9

Comments

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

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

Example

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.

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}]

Crossrefs

Cf. A000040, A001359, A006512, A236531.

Sequence in context: A378534 A366077 A367169 * A030205 A159817 A079532

Adjacent sequences: A236828 A236829 A236830 * A236832 A236833 A236834

Keyword

nonn

Author

Zhi-Wei Sun, Jan 31 2014

Status

approved