It’s been great to see all this work so I thought I’d post a few extra problems for people to try:
Let P_k be the set Of k-powerful numbers.
1) Enunerate P_3 as a_1<a_2<a_3 ...
Show that lim inf |a_{n+1}-a_{n}| = infinity
2) Let P be a polynomial with integer coefficients and at least 3 simple roots. Is it true that P(n) is
powerful only finitely often?
3) Pell’s Equation can be used to generate infinitely many pairs of powerful numbers. Can this or something similar be used to generate infinitely many pairs of powerful numbers inside a coprime arithmetic progression
I have a few more problems that I can post in due course