Dickman [Di30] showed the density of smooth $n$, with largest prime factor $<n^\alpha$, is $\rho(1/\alpha)$ where $\rho$ is the Dickman function.

Erdős also asked whether infinitely many such $n$ even exist, but Meza has observed that this follows immediately from Schinzel's result [Sc67b] that for infinitely many $n$ the largest prime factor of $n(n+1)$ is at most $n^{O(1/\log\log n)}$.

Erdős asked whether the events $P(n)<n^\alpha$ and $P(n+1)<(n+1)^\beta$ are independent, in the sense that the density of $n$ satisfying both conditions is equal to $\rho(1/\alpha)\rho(1/\beta)$.

Teräväinen [Te18] has proved the logarithmic density exists and is equal to $\rho(1/\alpha)\rho(1/\beta)$.

Wang [Wa21] has proved the density is $\rho(1/\alpha)\rho(1/\beta)$ assuming the Elliott-Halberstam conjecture for friable integers.

See also [370].

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