On the existence of odd, squarefree semiperfect numbers

Question

A positive integer $n$ is called square-free if no square divides it. Moreover, $n$ is called semiperfect (or pseudoperfect) if it is equal to the sum of some (or all) of its divisors. The smallest odd semiperfect number is $945$, and it has been proved that infinitely many odd semiperfect numbers exist. I have checked the first few odd semiperfect numbers, and noticed that none of them is square-free. Does there exist any odd semiperfect square-free number? If the answer is affirmative, what is the smallest one?

Answer 1

The ratio of the sum of divisors of $n$ (including $n$ itself) to $n$ must be at least than $2$ for a semi perfect number, and if $n$ is square-free this ratio is the product of $(p+1)/p$ for all its prime factors $p$. The smallest set of odd prime factors where the product formula is large enough is $\{3,5,7,11,13\}$ leading to

$15015=3×5×7×11×13=\text{ sum of its divisors except }15015,1155,1001,55,15$.