Sequence of 9 Consecutive Integers each with 48 Divisors
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Theorem
The $9$ integers beginning $17 \, 796 \, 126 \, 877 \, 482 \, 329 \, 126 \, 044$ each has $48$ divisors.
Proof
In the below, $\sigma_0$ denotes the divisor count function.
| \(\ds \map {\sigma_0} {17 \, 796 \, 126 \, 877 \, 482 \, 329 \, 126 \, 044}\) | \(=\) | \(\ds 48\) | $\sigma_0$ of $17 \, 796 \, 126 \, 877 \, 482 \, 329 \, 126 \, 044$ | |||||||||||
| \(\ds \map {\sigma_0} {17 \, 796 \, 126 \, 877 \, 482 \, 329 \, 126 \, 045}\) | \(=\) | \(\ds 48\) | $\sigma_0$ of $17 \, 796 \, 126 \, 877 \, 482 \, 329 \, 126 \, 045$ | |||||||||||
| \(\ds \map {\sigma_0} {17 \, 796 \, 126 \, 877 \, 482 \, 329 \, 126 \, 046}\) | \(=\) | \(\ds 48\) | $\sigma_0$ of $17 \, 796 \, 126 \, 877 \, 482 \, 329 \, 126 \, 046$ | |||||||||||
| \(\ds \map {\sigma_0} {17 \, 796 \, 126 \, 877 \, 482 \, 329 \, 126 \, 047}\) | \(=\) | \(\ds 48\) | $\sigma_0$ of $17 \, 796 \, 126 \, 877 \, 482 \, 329 \, 126 \, 047$ | |||||||||||
| \(\ds \map {\sigma_0} {17 \, 796 \, 126 \, 877 \, 482 \, 329 \, 126 \, 048}\) | \(=\) | \(\ds 48\) | $\sigma_0$ of $17 \, 796 \, 126 \, 877 \, 482 \, 329 \, 126 \, 048$ | |||||||||||
| \(\ds \map {\sigma_0} {17 \, 796 \, 126 \, 877 \, 482 \, 329 \, 126 \, 049}\) | \(=\) | \(\ds 48\) | $\sigma_0$ of $17 \, 796 \, 126 \, 877 \, 482 \, 329 \, 126 \, 049$ | |||||||||||
| \(\ds \map {\sigma_0} {17 \, 796 \, 126 \, 877 \, 482 \, 329 \, 126 \, 050}\) | \(=\) | \(\ds 48\) | $\sigma_0$ of $17 \, 796 \, 126 \, 877 \, 482 \, 329 \, 126 \, 050$ | |||||||||||
| \(\ds \map {\sigma_0} {17 \, 796 \, 126 \, 877 \, 482 \, 329 \, 126 \, 051}\) | \(=\) | \(\ds 48\) | $\sigma_0$ of $17 \, 796 \, 126 \, 877 \, 482 \, 329 \, 126 \, 051$ | |||||||||||
| \(\ds \map {\sigma_0} {17 \, 796 \, 126 \, 877 \, 482 \, 329 \, 126 \, 052}\) | \(=\) | \(\ds 48\) | $\sigma_0$ of $17 \, 796 \, 126 \, 877 \, 482 \, 329 \, 126 \, 052$ |
but then:
| \(\ds \map {\sigma_0} {17 \, 796 \, 126 \, 877 \, 482 \, 329 \, 126 \, 043}\) | \(=\) | \(\ds 32\) | $\sigma_0$ of $17 \, 796 \, 126 \, 877 \, 482 \, 329 \, 126 \, 043$ | |||||||||||
| \(\ds \map {\sigma_0} {17 \, 796 \, 126 \, 877 \, 482 \, 329 \, 126 \, 053}\) | \(=\) | \(\ds 8\) | $\sigma_0$ of $17 \, 796 \, 126 \, 877 \, 482 \, 329 \, 126 \, 053$ |
$\blacksquare$
Sources
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $17,796,126,877,482,329,126,044$