# Continued aliquot sums

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 🙂!

• I forget to add that I would also like to know under what conditions this extension of the aliquot sum concept is a local maximum, as was the case of the number 12 giving the result of 30. Jul 1 at 12:25

I'll answer one piece of the question. The sequence is listed at the OEIS, at https://oeis.org/A255242
Programs in Maple and in Mathematica are given for calculating terms in the sequence. No publications are referenced.
&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&
I think this answers all the questions in the original post, although one may not be happy with the answers.

1. "Is there an official term for this kind of number in extant mathematical journals?" No, at any rate none known to the contributors to the OEIS.
2. "Has this subject been researched before, and by whom?" It has been researched (or at any rate computed up to some limit) by those who submitted it to the OEIS, but, so far as they are aware, not by anyone else.
3. "What is the general formula for such numbers?" No such formula is known, at any rate, not to the OEIS contributors.
4. "Has a computer program been written that computes these numbers?" Yes, two such programs are given at the OEIS page.
5. "is this sequence already listed in the OEIS?" Yes, as indicated in the original part of this answer.
• Thanks for the answer! Has anyone researched the question of what kinds of compound numbers yield particularly large numbers of this type? Jul 1 at 12:28
• Sorry, all I know is what's written at the OEIS entry, which as I wrote doesn't reference any publications. It seems plausible that numbers $n$ for which the ordinary sum of divisors is large would also give rise to large value of the repeated function. Oh, the function is tabulated out to $10,000$ at the OEIS, that should get you started. Jul 1 at 12:33
• I have reason to believe that numbers that score high in this regard are always divisible by corresponding primorials, but that the smaller primes are taken to progressively higher powers. However, I do not yet have a formal proof for the exact formula, and it is still a conjecture on my part. Jul 1 at 12:34