2520 equals Sum of 4 Divisors in 6 Ways

Theorem
The number $2520$ can be expressed as the sum of $4$ of its divisors in $6$ different ways:

This is the maximum possible number of ways it is possible to express an integer as the sum of $4$ of its divisors.

Proof
We apply 1 can only be expressed as a sum of 4 reciprocals of unique positive integers in 6 ways:

Find the maximum powers of the primes in each equation, and choose the largest that appears:
 * $2^1 \times 3^1 \times 7^1$
 * $2^3 \times 3^1$
 * $2^1 \times 3^2$
 * $2^1 \times 3^1 \times 5^1$
 * $2^2 \times 5^1$
 * $2^2 \times 3^1$

Therefore the smallest number would be:
 * $2^3 \times 3^2 \times 5^1 \times 7^1 = 2520$