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}.

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, 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, 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

Offset: 1

Author

Zhi-Wei Sun, Apr 24 2015

Keywords

• nonn

Comments

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}.

Clearly, the conjecture implies the Twin Prime Conjecture.

Examples

			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.
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.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
• Zhi-Wei Sun, Natural numbers represented by floor(x^2/a) + floor(y^2/b) + floor(z^2/c), arXiv:1504.01608 [math.NT], 2015.

Crossrefs

Cf. A014574, A256555.

Programs

Mathematica

TQ[n_]:=PrimeQ[3n-1]&&PrimeQ[3n+1]
PQ[n_]:=TQ[3*n]||TQ[3*n+1]||TQ[3n+2]
Do[m=0;Do[If[PQ[x]&&PQ[n-x],m=m+1],{x,0,(n-1)/2}];
Print[n," ",m];Label[aa];Continue,{n,1,100}]