Sequence of 4 Consecutive Integers with Equal Number of Divisors

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.

$242, 243, 244, 245, 3655, 3656, 3657, 3658, 4503, 4504, 4505, 4506, \ldots$

$\ds \map {\sigma_0} {242}$

$=$

$\ds 6$

$\ds \map {\sigma_0} {243}$

$=$

$\ds 6$

$\ds \map {\sigma_0} {244}$

$=$

$\ds 6$

$\ds \map {\sigma_0} {245}$

$=$

$\ds 6$

$\ds \map {\sigma_0} {3655}$

$=$

$\ds 8$

$\ds \map {\sigma_0} {3656}$

$=$

$\ds 8$

$\ds \map {\sigma_0} {3657}$

$=$

$\ds 8$

$\ds \map {\sigma_0} {3658}$

$=$

$\ds 8$

$\blacksquare$