Sequence of 4 Consecutive Integers with Equal Number of Divisors
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Theorem
The following sequence of integers are sets of $4$ consecutive integers which all have the same number of divisors:
- $\map {\sigma_0} m = \map {\sigma_0} {m + 1} = \map {\sigma_0} {m + 2} = \map {\sigma_0} {m + 3}$
where $\map {\sigma_0} n$ denotes the divisor count function.
- $242, 243, 244, 245, 3655, 3656, 3657, 3658, 4503, 4504, 4505, 4506, \ldots$
This sequence is A039665 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).
Proof
\(\ds \map {\sigma_0} {242}\) | \(=\) | \(\ds 6\) | $\sigma_0$ of $242$ | |||||||||||
\(\ds \map {\sigma_0} {243}\) | \(=\) | \(\ds 6\) | $\sigma_0$ of $243$ | |||||||||||
\(\ds \map {\sigma_0} {244}\) | \(=\) | \(\ds 6\) | $\sigma_0$ of $244$ | |||||||||||
\(\ds \map {\sigma_0} {245}\) | \(=\) | \(\ds 6\) | $\sigma_0$ of $245$ |
\(\ds \map {\sigma_0} {3655}\) | \(=\) | \(\ds 8\) | $\sigma_0$ of $3655$ | |||||||||||
\(\ds \map {\sigma_0} {3656}\) | \(=\) | \(\ds 8\) | $\sigma_0$ of $3656$ | |||||||||||
\(\ds \map {\sigma_0} {3657}\) | \(=\) | \(\ds 8\) | $\sigma_0$ of $3657$ | |||||||||||
\(\ds \map {\sigma_0} {3658}\) | \(=\) | \(\ds 8\) | $\sigma_0$ of $3658$ |
$\blacksquare$
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $242$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $242$