Aliquot sequences

# Aliquot sequences

Aliquot sequences arise in iterating the sum-of-divisors function s, which assigns to a positive integer the sum of its proper divisors (i.e., excluding the number itself). An aliquot sequence thus starts with a positive integer n, followed by s(n), then s(s(n)), etc.

The main open question in the area is given by the Catalan-Dickson conjecture, which states that for any starting value n one of two things happens: either the sequence ends in a cycle, or it reaches 1 after finitely many steps (and is said to terminate then).

Cycles of length 1 come from perfect numbers: 6 = 1 + 2 + 3 = s(6), 28 = 1 + 2 + 4 + 7 + 14 = s(28), etc. Cycles of length 2 are also called amicable numbers: 220, s(220)=284, s(284)=220. Longer cycles are also called sociable numbers.

Progress in numerical work to verify the Catalan-Dickson conjecture relies upon progress in integer factorization methods, as no better method is known for the determination of the divisors of a number than the obvious method using its prime factorization.

I became interested in aliquot sequences while testing the integer factorization algorithms in Magma. I have listed some of the numerical work I have done. Several other people are involved in a big effort to push our knowledge for small starting values.

## Older work on aliquot sequences

For all n up to 50000 I have computed terms of the aliquot sequence starting with n until it terminated, entered an aliquot cycle, or reached an entry with at least 80 decimal digits. The aim now is to continue until at least 90 digits (or termination) is reached. All this is ongoing work as part of a bigger effort by several people (see the aliquot page of Wolfgang Creyaufmüller, or the page of Juan Varona, or the aliquot page of Paul Zimmermann).

# Sequences with starting values up to 50000

Here is a summary of the status of my own computations.
• Not(-yet)-terminating aliquot sequences with starting values up to 10000, summarized in table 1. This includes the current status of each of these sequences, as established by the work of Juan Varona and Manuel Benito, see their page, and that of Paul Zimmermann, who holds the current records for the five starting values less than 1000 remaining (the `Lehmer five') for which it is not yet known whether the sequence terminates, as well as for the sequences starting with 1074 and 1134; see his aliquot page.
• Not(-yet)-terminating aliquot sequences with starting values
• from 10000 up to 20000, summarized in table 2;
• from 20000 up to 30000, summarized in table 3;
• from 30000 up to 40000, summarized in table 4;
• and for starting values between 40000 and 50000 in table 5.
Most entries in these tables have been pushed beyond 90 digits; I am working my way down the list. As far as I know these two tables represent the current state of the art.
For sequences starting beyond 50000 check the aliquot page of Paul Zimmermann.