# 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