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
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Apr 25 2015
STATUS
approved