Problem Statement
Let $t_k(n)$ denote the least $m$ such that\[n\mid m(m+1)(m+2)\cdots (m+k-1).\]Is it true that\[\sum_{n\leq x}t_2(n)\ll \frac{x^2}{(\log x)^c}\]for some $c>0$?
Is it true that, for $k\geq 2$,\[\sum_{n\leq x}t_{k+1}(n) =o\left(\sum_{n\leq x}t_k(n)\right)?\]
Is it true that, for $k\geq 2$,\[\sum_{n\leq x}t_{k+1}(n) =o\left(\sum_{n\leq x}t_k(n)\right)?\]
Categories:
Number Theory
Progress
In [ErGr80] they mention a conjecture of Erdős that the sum is $o(x^2)$. This was proved by Erdős and Hall [ErHa78], who proved that in fact\[\sum_{n\leq x}t_2(n)\ll \frac{\log\log\log x}{\log\log x}x^2.\]Erdős and Hall conjecture that the sum is $o(x^2/(\log x)^c)$ for any $c<\log 2$.Since $t_2(p)=p-1$ for prime $p$ it is trivial that\[\sum_{n\leq x}t_2(n)\gg \frac{x^2}{\log x}.\]Erdős and Hall [ErHa78] also note that $t_{n-1}(n!)=2$ and $t_{n-2}(n!)\ll n$, which $n=2^r$ shows is the best possible. They ask about the behaviour of $t_{n-3}(n!)$ and also ask ask whether, for infinitely many $n$,\[t_k(n!)< t_{k-1}(n!)-1\]for all $1\leq k<n$. They proved (with Selfridge) that this holds for $n=10$.
Source: erdosproblems.com/394 | Last verified: January 14, 2026