A database of aliquot cycles

Jan Otto Munch Pedersen used to have a database of aliquot cycles, Tables of Aliquot Cycles, available on line, at the URL http://amicable.homepage.dk/tables.htm. Unfortunately, it doesn't seem to be available any more, so I am providing most of the data from this database here. The data is current as of the last time Pedersen's database was updated, which was on Oct. 1, 2007.

The following files provide all of the data from Pedersen's database except for ordinary perfect numbers. In these files, the exponential aliquot cycles present in Pedersen's database have been changed into modified exponential aliquot cycles (defined here.) Each member of an exponential aliquot cycle must have the same greatest squarefree divisor; the exponential aliquot cycles were changed into modified exponential cycles by dividing each member of the cycle by this divisor. The original exponential cycle can be recovered by multiplying all elements of the modified exponential cycle together, taking the greatest squarefree divisor of this product, and multiplying each element of the cycle by this divisor.


Each file is in ASCII text format and formatted as a sequence of lines. A sample line is as follows:

"Pedersen~1997" i.u. "Pedersen~1999" e. 4 2^ 13.59.727.1297 13.59.992441 23.1609.21559 23.33049169

The fields in this line should be interpreted as follows:

  1. i. is a code for the type of aliquot cycle. It means that this cycle is an infinitary aliquot cycle.
  2. u. is another type code, in this case meaning that this cycle is also a unitary aliquot cycle.
  3. "Pedersen~1997" means that Pedersen, in 1997, discovered that this cycle is unitary and infinitary aliquot. (In the original database, this discoverer information was written as Pedersen 1997. In these files, spaces in the discoverer information have been changed to tildes.)
  4. e. is a type code meaning that this cycle is also a modified exponential cycle.
  5. "Pedersen~1999" means that Pedersen, in 1999, discovered that the cycle obtainable from this cycle as described in the last section is exponential.
  6. The next field, 4, is the length of the cycle.
  7. The following field, 2^, is the greatest common unitary divisor (g, say) of all elements of the cycle, written as a product of prime powers. Another way of putting this is that g is the product of all prime powers which occur with the same exponent in the factorizations of every element of the cycle. The empty product which is the factorization of 1 is written as 1.
  8. The remaining four fields, 13.59.727.1297, ... are the four elements of the cycle, each divided by g. Again, each quotient is written as a product of prime powers.

In Pedersen's database, this line would have been split into three cycles (one of each type), and represented by the following three sets of six lines:

I4 Pedersen 1997

U4 Pedersen 1997

E4 Pedersen 1999

The type codes are as follows:

Code used
in these files
Code used
by Pedersen
n.C (or no code)Ordinary (normal)
u+AUAugmented unitary
u-RUReduced unitary
i+AIAugmented infinitary
i-RIReduced infinitary
e.E(Reduced) exponential


The files are as follows:

File name Size MD5 Number of lines Number of cycles contained,
listed by type and length
aliquot-database-.1.txt 65 KB 0d3b1ef74311e382f24ab60b5353a54c 199 u.1: 5
i.1: 190
e.1: 12
aliquot-database-.2.txt.bz2 1.3 GB
6.9 GB
22728020 n.2: 11994387
u.2: 4911908
i.2: 11538100
e.2: 3089296
aliquot-database-.4.txt 1.7 MB ab9a32462b26afdade33f4c6b6046ed6 5518 n.4: 142
u.4: 191
i.4: 5034
e.4: 371
aliquot-database-.other.txt 44 KB 921476ac5baf13d5b98fd1d7174df342 187 n.: 10 (lengths 5–28)
u.: 24 (lengths 3, 5–65)
i.: 129 (lengths 3, 6–85)
e.: 38 (lengths 3, 5–61)
aliquot-database-+2.txt 121 KB 77afd633caaf66ec4a498aee3d710aac 1956 n+2: 1931
u+2: 27
i+2: 425
aliquot-database--2.txt 122 KB da0cff8e6223b20144783623461c7ee8 1983 n-2: 1946
u-2: 28
i-2: 427
aliquot-database-+4.txt 193 B 4dc438252e86c20c1bbce419efba53cb 2 n+4: 2
aliquot-database--8.txt 157 B a2800c7d33e675971a3fe07cfdb7b7c9 1 n-8: 1

David Moews (dmoews@fastmail.fm)

Last significant revision 4-II-2015