Smallest Numbers with 240 Divisors
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Theorem
The smallest integers with $240$ divisors are:
- $720 \, 720, 831 \, 600, 942 \, 480, 982 \, 800, 997 \, 920, \ldots$
Proof
In the below, $\map {\sigma_0} n$ denotes the divisor count function of $n$.
Then:
\(\ds \map {\sigma_0} {720 \, 720}\) | \(=\) | \(\ds 240\) | $\sigma_0$ of $720 \, 720$ | |||||||||||
\(\ds \map {\sigma_0} {831 \, 600}\) | \(=\) | \(\ds 240\) | $\sigma_0$ of $831 \, 600$ | |||||||||||
\(\ds \map {\sigma_0} {942 \, 480}\) | \(=\) | \(\ds 240\) | $\sigma_0$ of $942 \, 480$ | |||||||||||
\(\ds \map {\sigma_0} {982 \, 800}\) | \(=\) | \(\ds 240\) | $\sigma_0$ of $982 \, 800$ | |||||||||||
\(\ds \map {\sigma_0} {997 \, 920}\) | \(=\) | \(\ds 240\) | $\sigma_0$ of $997 \, 920$ |
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Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $240$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $240$