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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.

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  • $\begingroup$ Sequence/segment $\ 2,3,6,8,12,14,18\ $ appears here: oeis.org/…. This may (or may not) be accidental. I'll write a simple Perl program, it will expand the initial segments of $\ v_n\ $ and $\ U_n$. $\endgroup$
    – Wlod AA
    Oct 31, 2022 at 3:24
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    $\begingroup$ OEIS A001694 links to Paul T. Bateman and Emil Grosswald, On a theorem of Erdős and Szekeres, Illinois J. Math. 2:1 (1958), pp. 88-98, and notes that it gives $\textrm{POW}(x) = \frac{\zeta(\tfrac 32)}{\zeta(3)} x^{\tfrac12} + \frac{\zeta(\tfrac 23)}{\zeta(2)} x^{\tfrac13} + o(x^{\tfrac16})$ $\endgroup$ Oct 31, 2022 at 12:05

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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.

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