A237839 a(n) = |{0 < k <= n: q = |{p <= k*n: p and p + 2 are both prime}| and q + 2 are both prime}|.

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

Offset: 1

Author

Zhi-Wei Sun, Feb 14 2014

Keywords

• nonn

Comments

Conjecture: a(n) > 0 for all n > 3, and a(n) = 1 only for n = 5, 9, 25, 77, 104.

See also A237838 for a similar conjecture involving Sophie Germain primes.

Examples

			a(9) = 1 since {p <= 4*9: p and p + 2 are both prime} = {3, 5, 11, 17, 29} has cardinality 5 and {5, 7} is a twin prime pair.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..1100

Crossrefs

Cf. A001359, A006512, A237578, A237769, A237817, A237838.

Programs

Mathematica

TQ[n_]:=PrimeQ[n]&&PrimeQ[n+2]
tq[n_]:=Sum[If[PrimeQ[Prime[k]+2],1,0],{k,1,PrimePi[n]}]
a[n_]:=Sum[If[TQ[tq[k*n]],1,0],{k,1,n}]
Table[a[n],{n,1,80}]