Self-Replicating Numbers
Jelly, 11 bytes
1ẇⱮ×R$SƊ=¥#

Try it online!
Takes \$n\$ then \$m\$ on the command line and outputs the first \$m\$ \$n\$-order self-replicating numbers
How it works
1ẇⱮ×R$SƊ=¥# - Main link. Takes n on the left
         ¥  - Group the previous 2 links into a dyad f(k, n):
       Ɗ    -   Group the previous 3 links into a monad on k:
     $      -     Group the previous 2 links into a monad on k:
    R       -       Yield the range [1, 2, ..., k]
   ×        -       Multiply each element by k
  Ɱ         -     For each element i in [k, 2k, ..., k²]:
 ẇ          -       Is k a substring of i?
      S     -     Sum; Count the 1s
        =   -   Does this equal n?
1         # - Count up k = 1, 2, 3, ... until m such k return true under f(k, n)