cp's OEIS Frontend

A237817 Number of primes p < n such that r = |{q <= n-p: q and q + 2 are both prime}| and r + 2 are both prime.

%I A237817 #8 Apr 06 2014 22:21:45
%S A237817 0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,3,3,4,3,2,2,2,2,2,2,2,2,2,2,2,3,4,4,5,
%T A237817 5,5,5,5,5,6,6,8,7,6,6,5,5,5,5,5,5,6,6,5,5,5,5,5,5,4,4,4,4,3,3,4,4,4,
%U A237817 4,4,4,5,5,5,5,5,5,4,4,4
%N A237817 Number of primes p < n such that r = |{q <= n-p: q and q + 2 are both prime}| and r + 2 are both prime.
%C A237817 Conjecture: (i) a(n) > 0 for all n > 12.
%C A237817 (ii) For any integer n > 2, there is a prime p < n such that r = |{q <= n-p: q and q + 2 are both prime}| is a square.
%C A237817 See also A237815 for a similar conjecture involving Sophie Germain primes.
%H A237817 Zhi-Wei Sun, <a href="/A237817/b237817.txt">Table of n, a(n) for n = 1..10000</a>
%H A237817 Z.-W. Sun, <a href="http://arxiv.org/abs/1402.6641">Problems on combinatorial properties of primes</a>, arXiv:1402.6641, 2014
%e A237817 a(13) = 1 since {q <= 13 - 2: q and q + 2 are both prime} = {3, 5, 11} has cardinality 3, and {3, 3 + 2} is a twin prime pair.
%t A237817 TQ[n_]:=PrimeQ[n]&&PrimeQ[n+2]
%t A237817 sum[n_]:=Sum[If[PrimeQ[Prime[k]+2],1,0],{k,1,PrimePi[n]}]
%t A237817 a[n_]:=Sum[If[TQ[sum[n-Prime[k]]],1,0],{k,1,PrimePi[n-1]}]
%t A237817 Table[a[n],{n,1,80}]
%Y A237817 Cf. A000040, A000290, A001359, A006512, A237705, A237706, A237769, A237815.
%K A237817 nonn
%O A237817 1,14
%A A237817 _Zhi-Wei Sun_, Feb 13 2014