Although the answer to my question is probably implicit in the answers to the question asked here: Density of numbers having large prime divisors (formalizing heuristic probability argument), I can't extract it.

Problem: find decent bounds on the number of positive integers $n$, such that, for all primes $p$ dividing $n$, if $p^k$ exactly divides $n$, then $n > p^{k+1}$.

My idea for a first upper bound: if $n$ is divisible by a prime larger than $n^{\frac{1}{2}}$, it is immediately exluded that $n$ is of the above form, so the density can never be larger than $1 - \log{2}$

My idea for a first lower bound: if $n$ has two prime divisors between $n^{\frac{1}{3}}$ and $n^{\frac{1}{2}}$, then $n$ is of the above form. But I don't know the density of these numbers.

I am probably very happy with a (reference to) a proof/theorem that implies that we have a positive lower density, but asymptotics would be great.

EDIT (after the first response of GH, for which I'm thankful!): assume $n$ lies in some moduloclass, say $a \pmod{b}$. Can we still show a positive lower density, whatever the values of $a$ and $b$?