A257317 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.

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, 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, 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

Offset

1,3

Comments

Conjecture: a(n) > 0 for all n > 0.

Clearly, this conjecture implies the Twin Prime Conjecture. Note that a(n) does not exceed A256707(n).

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.

Example

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.
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.

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[Mod[x(n-x), 2]==0&&PQ[x]&&PQ[n-x], m=m+1], {x, 0, (n-1)/2}];
Print[n, " ", m]; Label[aa]; Continue, {n, 1, 100}]

Crossrefs

Cf. A014574, A256707, A257121.

Sequence in context: A274828 A364136 A257474 * A163376 A261913 A088601

Adjacent sequences: A257314 A257315 A257316 * A257318 A257319 A257320

Keyword

nonn

Author

Zhi-Wei Sun, Apr 25 2015

Status

approved