Yesterday I met professor Halfbrain in the city. The professor looked tired and somewhat exhausted. He told me that he had spent his nights and days with adding up digits of divisors of positive integers. For instance the integer $n=12$ has the six divisors $1,2,3,4,6,12$, and the sum of the digits of these six divisors is $1+2+3+4+6+1+2=19$. The professor denoted the sum of the digits of all divisors of $n$ by $SDD(n)$, so that in particular $SDD(12)=19$. The professor started with some arbitrary integer $n$, computed $SDD(n)$, then computed $SDD(SDD(n))$, then $SDD(SDD(SDD(n)))$, and so on.
Professor Halfbrain's theorem: If we start with an arbitrary positive integer $n\ge2$ and iteratively compute the sum SDD of the digits of all divisors, then we eventually reach the integer $15$.
For example, let us start with $n=4$. Then the next integer is $SDD(4)=1+2+4=7$. The next integer is $SDD(7)=1+7=8$. The next integer is $SDD(8)=1+2+4+8=15$. Then we are stuck, as $SDD(15)=1+3+5+1+5=15$.
Is the professor's theorem really correct, or has the professor once again made one of his phenomenal mathematical blunders?