Problem 461

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Statement. 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 $n$)?

Status. Open.

Source. erdosproblems.com/461, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #461, https://www.erdosproblems.com/461.

Formalization. None recorded.

Progress

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