Consecutive non-powerful integers

Question

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.

Answer 1

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, but I would be surprised if it was not $\sim2\sqrt{v_n}$ as well.