We say that a natural number $n$ has triangular divisors if it has at least one triplet of divisors $n = d_1d_2d_3$, $1 \le d_1 \le d_2 \le d_3$, such that $d_1,d_2$ and $d_3$ form the sides of a triangle (non degenerate)
E.g.: $60$ has triangular divisors because $60 = 3.4.5$ and $3,4,5$ form a triangle. Note that another triplet of divisors of $60 = 1.4.15$ does not form a triangle but because of the triplet $3,4,5$ the number $60$ qualifies a number with triangular divisors. On the other the number $10$ does not have any triplet of triangular divisors.
I found the following conjectures experimentally. Can they be proved or disproved?
Weak conjecture: Every integer $\ge 8$ which has triangular divisors can be written as the sum of two integers both of which has triangular divisors.
Strong conjecture: Every integer $\ge 8$ except $11, 14, 15,23, 38, 47, 55, 71, 103, 113$ and $311$ can be written as the sum of two integers both of which has triangular divisors.
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