# Consecutive non-powerful integers

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.

• 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 $\ U_n$. Oct 31, 2022 at 3:24
• 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 $\textrm{POW}(x) = \frac{\zeta(\tfrac 32)}{\zeta(3)} x^{\tfrac12} + \frac{\zeta(\tfrac 23)}{\zeta(2)} x^{\tfrac13} + o(x^{\tfrac16})$ Oct 31, 2022 at 12:05

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.