cp's OEIS Frontend

A256707 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 primes}.

%I A256707 #26 Feb 28 2023 12:00:17
%S A256707 1,1,2,2,2,2,2,2,2,2,2,3,2,3,3,3,3,3,3,3,3,4,5,4,3,5,5,5,3,3,5,5,5,5,
%T A256707 3,5,6,5,2,3,5,6,2,1,3,7,4,3,4,5,5,5,3,5,3,4,3,3,5,4,3,3,4,5,2,5,4,5,
%U A256707 6,4,5,6,5,7,3,4,5,4,6,3,3,4,4,5,3,3,2,5,5,2,4,6,7,7,4,6,6,6,6,3
%N A256707 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 primes}.
%C A256707 Conjecture: a(n) > 0 for all n > 0. Moreover, for any integer m > 10 every positive integer can be written as the sum of two distinct elements of the set {floor(x/m): x-1 and x+1 are twin prime}.
%C A256707 Clearly, the conjecture implies the Twin Prime Conjecture.
%H A256707 Zhi-Wei Sun, <a href="/A256707/b256707.txt">Table of n, a(n) for n = 1..10000</a>
%H A256707 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 A256707 a(44) = 1 since 44 = 6 + 38 = floor(20/3) + floor(116/3) with {3*20-1,3*20+1} = {59,61} and {3*116-1,3*116+1} = {347,349} twin prime pairs.
%e A256707 a(108) = 1 since 108 = 16 + 92 = floor(50/3) + floor(276/3) with {3*50-1,3*50+1} = {149,151} and {3*276-1,3*276+1} = {827,829} twin prime pairs.
%t A256707 TQ[n_]:=PrimeQ[3n-1]&&PrimeQ[3n+1]
%t A256707 PQ[n_]:=TQ[3*n]||TQ[3*n+1]||TQ[3n+2]
%t A256707 Do[m=0;Do[If[PQ[x]&&PQ[n-x],m=m+1],{x,0,(n-1)/2}];
%t A256707 Print[n," ",m];Label[aa];Continue,{n,1,100}]
%Y A256707 Cf. A014574, A256555.
%K A256707 nonn
%O A256707 1,3
%A A256707 _Zhi-Wei Sun_, Apr 24 2015