A237839 - OEIS

%I #9 Feb 14 2014 03:29:16

%S 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,

%T 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,

%U 5,3,5,8,4,3,3,4,1,3,4,3

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

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

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

%H Zhi-Wei Sun, <a href="/A237839/b237839.txt">Table of n, a(n) for n = 1..1100</a>

%e 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.

%t TQ[n_]:=PrimeQ[n]&&PrimeQ[n+2]

%t tq[n_]:=Sum[If[PrimeQ[Prime[k]+2],1,0],{k,1,PrimePi[n]}]

%t a[n_]:=Sum[If[TQ[tq[k*n]],1,0],{k,1,n}]

%t Table[a[n],{n,1,80}]

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

%K nonn

%O 1,4

%A _Zhi-Wei Sun_, Feb 14 2014