15. If a four-digit number $n=\overline{a b c d}$ has digits $a, b, c, d$ such that any three of these digits can form the lengths of the sides of a triangle, then $n$ is called a "four-digit triangular number". Find the total number of four-digit triangular numbers.
15. Call $(a, b, c, d)$ a digital group of $n$, then
$$
a, b, c, d \in M=\{1,2, \cdots, 9\} \text {. }
$$
(1) When the digital group contains only one value, i.e., $(a, a, a, a)(a=1,2, \cdots, 9)$, there are 9 values of $n$.
(2) When the digital group contains exactly two values $a, b(a>b)$.
(i) The digital group is of type $(a, a, a, b)$, then any three digits can form a triangle. For each $a \in\{2,3, \cdots, 9\}, b$ can take $a-1$ values, so the number of digital groups is $\sum_{a=2}^{9}(a-1)=36$. For each group $(a, a, a, b), b$ has 4 placement methods, thus, there are $36 \times 4=144$ such $n$.
(ii) The digital group is of type $(a, b, b, b)$, according to the triangle formation condition, $b < a < 2b$.
(i) The digital group is of type $(a, b, c, c)$, according to the triangle formation condition, then $c < b < a < 2c$, there are 14 such $(a, b, c, c)$, each group has $A_4^2 = 12$ placement methods, thus, there are $14 \times 12 = 168$ such $n$.
(ii) The digital group is of type $(a, b, b, c)$, $c < b < a < b+c$, this condition is equivalent to the number of ways to form a triangle by taking 3 different numbers from $M=\{1,2, \cdots, 9\}$, there are 34 groups, each group has $A_4^2 = 12$ placement methods, thus, there are $34 \times 12 = 408$ such $n$.
(iii) The digital group is of type $(a, a, b, c)$, $c < b < a < b+c$, same as case (ii), there are $34 A_4^2 = 408$ such $n$ values.
In total, there are $168 + 408 + 408 = 984$ such $n$.
(4) $a, b, c, d$ are all different, then $d < c < b < a < c+d$, there are 16 such $a, b, c, d$, each group has $4!$ arrangements, thus, there are $16 \times 4! = 384$ such $n$ values.
In summary, the total number of four-digit triangular numbers $n$ is
$$
9 + 304 + 984 + 384 = 1681 \text{ (values). }
$$
(Provided by Tao Pingsheng)