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%I #15 Apr 25 2015 11:45:49
%S 1,1,2,1,2,2,2,2,2,2,2,2,2,2,3,2,3,2,3,3,3,3,5,3,3,3,5,4,3,3,5,3,5,4,
%T 3,3,6,5,2,2,5,5,2,1,3,5,4,3,4,5,5,3,3,4,3,3,3,3,5,4,3,2,4,4,2,3,4,5,
%U 6,4,5,4,5,4,3,2,5,3,6,3,3,2,4,3,3,2,2,3,5,2,4,4,7,4,4,4,6,4,6,3
%N Number of unordered ways to write n as the sum of two distinct elements of the set {floor(x/3): 3*x-1 and 3*x+1 are twin prime} one of which is even.
%C Conjecture: a(n) > 0 for all n > 0.
%C Clearly, this conjecture implies the Twin Prime Conjecture. Note that a(n) does not exceed A256707(n).
%H Zhi-Wei Sun, <a href="/A257317/b257317.txt">Table of n, a(n) for n = 1..10000</a>
%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1504.01608">Natural numbers represented by floor(x^2/a) + floor(y^2/b) + floor(z^2/c)</a>, arXiv:1504.01608 [math.NT], 2015.
%e a(4) = 1 since 4 = 0 + 4 = floor(2/3) + floor(14/3) with 0 or 4 even, and {3*2-1,3*2+1} = {5,7} and {3*14-1,3*14+1} = {41,43} twin prime pairs.
%e a(108) = 1 since 108 = 16 + 92 = floor(50/3) + floor(276/3) with 16 or 92 even, and {3*50-1,3*50+1} = {149,151} and {3*276-1,3*276+1} = {827,829} twin prime pairs.
%t TQ[n_]:=PrimeQ[3n-1]&&PrimeQ[3n+1]
%t PQ[n_]:=TQ[3*n]||TQ[3*n+1]||TQ[3n+2]
%t Do[m=0;Do[If[Mod[x(n-x),2]==0&&PQ[x]&&PQ[n-x],m=m+1],{x,0,(n-1)/2}];
%t Print[n," ",m];Label[aa];Continue,{n,1,100}]
%Y Cf. A014574, A256707, A257121.
%K nonn
%O 1,3
%A _Zhi-Wei Sun_, Apr 25 2015
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