240

The maximum number of divisors for any number less than 1,000,000. 5 numbers less than a million have 240 divisors:

720720 = 2⁴3²5•7•11•13

831600 = 2⁴3³5²•7•11

942480 = 2⁴3²•5•7•11•17

982800 = 2⁴3³5²•7•13

997920 = 2⁵3⁴•5•7•11

240 = 1•1•2•3•5•8 (the product of the first 6 Fibonacci numbers)

240 + 1 and 240 - 1 are twin primes.