# ALIQUOT SEQUENCES

Other aliquot pages: Wieb Bosma, Christophe Clavier, Wolfgang Creyaufmueller, Clifford Stern, Juan Luis Varona, www.rieselprime.de, MersenneForum, Markus Tervooren's Factoring Database, AllSeq, Jean-Luc Garambois [in french].

News:

• 25 March 2017: I stop extending the 276 sequence, after finding no factor by ECM of the c201 (tested up to 60 digits) at index 2051
• 11 September 2009: thanks to Karsten Bonath, the factors I found for sequences 276, 552, 564, 660, 966, 1074, 1134 and 204848 are now available on Markus Tervooren's Factoring Database.
• The aliquot sequence starting from n is defined as follows: let σ(n) be the sum of divisors of n, then one simply computes f(n)=σ(n)-n, and one iterates. For example, if we start from 12, whose divisors are 1, 2, 3, 4, 6, 12, then σ(12)=1+2+3+4+6+12=28 and therefore f(12)=16. Then f(16)=15, f(15)=9, f(9)=4, f(4)=3, f(3)=1, f(1)=0, f(0)=0, and one then loops on 0. One can also loop on perfect numbers, i.e. numbers such that f(n)=n, for example n=6. An open question asked by M. E. Catalan in 1888 is whether any aliquot sequence eventually reaches 1, a perfect number, or a cycle of amicable or sociable numbers. Lehmer tried to investigate with the computer the sequences of starting value less than 1000, and found that all terminate, except perhaps for n=276, 552, 564, 660 and 966.

Continuing work of Wolfgang Creyaufmueller, with the help of Sam Wagstaff, Arjen Lenstra, Peter Montgomery and Ryan Propper, I have extended the ``Lehmer five'' sequences, together with the sequences starting from 1074, 1134, 19560 and 204828 (sequence 4788 is also interesting):

 starting value current index comment 276 last stopped at index 2051 with c201 552 last stopped at index 1142 with c185 (t55) 564 last stopped at index 3486 with c197 (t60) 660 last stopped at index 1008 with c182 (t60) 966 last stopped at index 1035 with c192 (t55) 1074 last stopped at index 2194 with c181 (t55) 1134 last stopped at index 3839 with c176 19560 last stopped at index 590 with c203 (t60) 204828 last stopped at index 4888 with c189 (t60)
Note: t60 means "tested up to 60 digits by ECM".

## Tools and links for aliquot sequences.

• The aliq.c C program enables one to decode and check an aliquot sequence encoded with the following format (files xxx.fmt in the above table).
• The aliq2.c is an extended version of the above C program, which tries to factor composites with the ecm library.
• The Amicable pairs list from Sergei Chernykh
• The YAFU project