My Algorithm

• Keep a base of controlling vectors for each prime number.
• Keep $n_vec - the vector starting with n[sq] that has to be composed out of the base.
• As we encounter the next integer $i -
• 1. Get its squaring factors
  2. Check if it's not square or prime (which are useless for us)
  3. Stair-shape it, by multiplying it with the controlling vectors of its factors
  4. Possibly assign it as the controlling vector of the minimal (ID-wise) prime in its stair shape version, and canonize the rest of the base accordingly.
  5. Try to canonize $n_vec.

• Once $n_vec becomes 0, we stop and return.