Properties of Family of 333,667 and Related Numbers

Theorem
This page reports on certain properties, difficult to classify, of the number $333 \, 667$, and patterns arising.

The square of any number consisting of:
 * a string of $3$s

followed by:
 * a string of $6$s

followed by:
 * a single $7$

has its digits all in an increasing sequence:

The same is true of numbers of the following form: