cp's OEIS Frontend

A236566 Number of ordered ways to write 2*n = p + q with p, q and prime(p + 2) + 2 all prime.

%I A236566 #15 Jan 29 2014 10:33:02
%S A236566 0,0,1,2,2,1,2,3,2,1,3,2,1,2,1,1,4,2,1,2,3,3,4,5,4,4,5,2,4,4,3,5,3,1,
%T A236566 5,6,4,3,6,2,4,8,4,3,6,3,4,3,3,4,5,4,3,6,6,5,8,3,4,7,2,3,5,2,4,4,3,3,
%U A236566 6,5,4,6,3,4,7,3,5,4,2,4,4,1,2,7,4,2,5,3,5,6,4,4,4,2,3,4,4,4,5,2
%N A236566 Number of ordered ways to write 2*n = p + q with p, q and prime(p + 2) + 2 all prime.
%C A236566 Conjecture: (i) a(n) > 0 for all n > 2.
%C A236566 (ii) If n > 30, then 2*n + 1 can be written as 2*p + q with p, q and prime(p + 2) + 2 all prime.
%C A236566 Part (i) implies both the Goldbach conjecture and the twin prime conjecture. If all primes p with prime(p + 2) + 2 are smaller than an even number N > 2, then for any such a prime p the number N! + N - p is in the interval (N!, N! + N) and hence not prime.
%C A236566 Similarly, part (ii) implies both Lemoine's conjecture (cf. A046927) and the twin prime conjecture.
%C A236566 We have verified part (i) of the conjecture for n up to 2*10^8.
%H A236566 Zhi-Wei Sun, <a href="/A236566/b236566.txt">Table of n, a(n) for n = 1..10000</a>
%e A236566 a(10) = 1 since 2*10 = 3 + 17 with 3, 17 and prime(3 + 2) + 2 = 11 + 2 = 13 all prime.
%e A236566 a(589) = 1 since 2*589 = 577 + 601 with 577, 601 and prime(577 + 2) + 2 = 4229 + 2 = 4231 all prime.
%t A236566 p[m_]:=PrimeQ[Prime[m+2]+2]
%t A236566 a[n_]:=Sum[If[p[Prime[k]]&&PrimeQ[2n-Prime[k]],1,0],{k,1,PrimePi[2n-1]}]
%t A236566 Table[a[n],{n,1,100}]
%Y A236566 Cf. A000040, A001359, A002372, A002375, A006512, A046927, A236531.
%K A236566 nonn
%O A236566 1,4
%A A236566 _Zhi-Wei Sun_, Jan 28 2014