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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. 4
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 (list; graph; refs; listen; history; text; internal format)
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: A130634 A274828 A257474 * A163376 A261913 A088601

Adjacent sequences:  A257314 A257315 A257316 * A257318 A257319 A257320

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Apr 25 2015

STATUS

approved

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Last modified January 26 10:34 EST 2021. Contains 340438 sequences. (Running on oeis4.)