What happens if one takes the aliquot sum of an integer and then repeats the process so that one takes the aliquot sum of all of those factors that were not reduced to the number 1 on the previous round, and then continues in like fashion so that one has a branching diagram of all the "factors of factors of factors..." and so on, and then one sums them all?
For example, for primes there will be no difference since the aliquot sum of primes is $1$ to begin with. For numbers consisting of just two primes, however, this "continued aliquot sum" will always be $2$ more than the "primitive" aliquot sum, because the aliquot sums of each of the two primes will each have an aliquot sum of $1$.
For squares of a prime the continued aliquot sum will always be one more than the primitive aliquot sum as the aliquot sum of the prime has to be taken again, giving an additional $1$. For the number $8$ we get $1+2+4+(1)+(1+2)+(1)=12$. For the number $12$ we get $1+2+3+4+6+(1)+(1)+(1+2)+(1+2+3)+(1)+(1+1)=30$.
I have several questions here: Is there an official term for this kind of number in extant mathematical journals? Has this subject been researched before, and by whom? What is the general formula for such numbers? Has a computer program been written that computes these numbers? If we take the sequence of all of these so-called "continuous aliquot sums" of numbers, is this sequence already listed in the OEIS? Thank in advance for any answers, even partial 🙂!