Pair of sequences $\ v_n\ $ and $\ U_n\ $ of integers start as in the following table:

[\begin{array}{rrrrrrrrrr} n= & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & \ldots \\ v_n= & 0 & 2 & 5 & 10 & 17 & 37 & 50 & 82 & \ldots \\ U_n= & 0 & 2 & 3 & 6 & 8 & 12 & 14 & 18 & \ldots \end{array}]

These two sequences are defined as follows:

- $\ v_0=U_0=0;$
- $\ v_n\in\mathbb N\ $ is the smallest natural number such that
**none**of the consecutive $\,\ U_{n-1}\!+\!1\,\ $ integers $\ v_n\ \ldots\ v_n\!+\!U_{n-1}\ $ is powerful; - $\ U_n\ $ is the smallest natural number such $\ v_n+U_n\ $
**is**powerful.

Thus, we are looking at the ever longer maximal sequences of consecutive non-powerful sequences. One would like to know the behavior of these sequences:

**Question**: what are reasonable (as exact as possible, and easily computable) lower and upper bounds for terms $\ v_n\ $ and $\ U_n,\ $ and their asymptotic behavior?

Knowing roughly the number of powerful initegers $\ POW(x)\ $ that do not exceed $\ x\ $ (for every positive $\ x\in\mathbb R),\ $ we may deduce the average behavior of these sequences of non-powerful integers; the still harder challenge would be deducing the more delicate but consistent deviations from the regular statistical behavior.