Problem 461

Erdős Problem #461: declared status 'open'. Formalized: no. Let $s_t(n)$ be the $t$-smooth component of $n$ - that is, the product of all primes $p$ (with multiplicity) dividing $n$ such that $p<t$. Let $f(n,t)$ count the number of distinct possible values for $s_t(m)$ for $m\in [n+1,n+t]$. Is it true that$$f(n,t)\gg t$$(uniformly, for all $t$ and $n$)? Prize: no. Tags: number theory, primes.