For the positive integer $n$, let $\langle nangle$ denote the sum of all the positive divisors of $n$ with the exception of $n$ itself. For example, $\langle 4angle=1+2=3$ and $\langle 12 angle =1+2+3+4+6=16$. What is $\langle\langle\langle 6angleangleangle$?