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For every fixed sufficiently small $\delta>0$ and all sufficiently large $n$, at least $(1-2\delta)n$ integers in $[n]$ are $n^{1-\delta}$-powersmooth.

In particular, for fixed sufficiently small $\varepsilon>0$ and $Q=n^{1-\varepsilon}/2$, at least $(1-4\varepsilon)n$ integers in $[n]$ are $Q$-powersmooth for sufficiently large $n$.

Source: published PDF, Lemma 5, p. 9. The first-minimal-power grouping makes the printed exception estimate explicit; this step is valid in the source. The second deduction uses $\delta=2\varepsilon$, rather than silently discarding the factor $1/2$ in $Q$.

Bears on. Problem 297.

Proof

Put $t=n^{1-\delta}$. The near-one Dickman estimate gives $(1+\log(1-\delta)+o_\delta(1))n$ $t$-smooth integers. For $0<\delta\le1/4$, $-\log(1-\delta)\le\delta/(1-\delta)\le4\delta/3$. There are consequently at least $(1-3\delta/2)n$ such integers for large $n$.

If one is not $t$-powersmooth, some prime $p\le t$ has a power dividing it that exceeds $t$. For each such $p$, choose the least exponent $a(p)$ with $p^{a(p)}>t$. All exceptional integers associated with $p$ are divisible by this single minimal power, including those having higher powers. Their number is at most $\lfloor n/p^{a(p)}\rfloor\le\lfloor n/t\rfloor$. The union bound over the primes gives at most

$$\pi(t)\lfloor n/t\rfloor\le n\,\pi(t)/t=o(n)$$

exceptions, by the prime number theorem. For large $n$ this is at most $\delta n/2$. Subtracting proves the first assertion.

For the second, $Q\ge n^{1-2\varepsilon}$ once $n^\varepsilon\ge2$. An integer that is $n^{1-2\varepsilon}$-powersmooth is therefore $Q$-powersmooth. Apply the first assertion with $\delta=2\varepsilon$.