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.



Consecutive non-powerful integers
Infinitely often (and with positive density, as proved by Shiu) there are no powerful numbers between $n^2+1$ and $(n+1)^2-1$. Hence the maximal gap below $x$ is infinitely often of size $\sim2\sqrt x$. (This is more than can be deduced from the Erdős & Szekeres result, or the Bateman & Grosswald result, since $\zeta(3/2)/\zeta(3)>2.$)


$U\_n$ is largest if $v\_n$ is a square and the above interval is empty. So
$$
U\_n \le (\sqrt{v\_n}+1)^2 - v\_n = 2\sqrt{v\_n}+1
$$
with equality holding in that case. Deducing a lower bound is probably tantamount to finding gaps in [A336175](https://oeis.org/A336175), but I would be surprised if it was not $\sim2\sqrt{v\_n}$ as well.


