Erdős Problem #461

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 $n$)?

#461: [ErGr80,p.92]

number theory | primes

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Erdős and Graham report they can show\[f(n,t) \gg \frac{t}{\log t}.\]

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This page was last edited 28 October 2025.

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