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.

Format

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 2^4.3.7.11.17 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, 2^4.3.7.11.17, 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
45441288228336=2^4*3*7*11*17*13*59*727*1297
47827859583504=2^4*3*7*11*17*13*59*992441
50129500421616=2^4*3*7*11*17*23*1609*21559
47760543891984=2^4*3*7*11*17*23*33049169

U4 Pedersen 1997
45441288228336=2^4*3*7*11*17*13*59*727*1297
47827859583504=2^4*3*7*11*17*13*59*992441
50129500421616=2^4*3*7*11*17*23*1609*21559
47760543891984=2^4*3*7*11*17*23*33049169

E4 Pedersen 1999
6754434846980278000063949240990877631874113595424=2^5*3^2*7^2*11^2*13^2*17^2*23*59^2*727^2*1297^2*1609*21559*992441*33049169
7109177006690875969899287052947525278263326087136=2^5*3^2*7^2*11^2*13^2*17^2*23*59^2*727*1297*1609*21559*992441^2*33049169
7451295016287318915996865686292912738108671526944=2^5*3^2*7^2*11^2*13*17^2*23^2*59*727*1297*1609^2*21559^2*992441*33049169
7099171140434035641787234987266949507863478335456=2^5*3^2*7^2*11^2*13*17^2*23^2*59*727*1297*1609*21559*992441*33049169^2

The type codes are as follows:

Files

The files are as follows:

David Moews ([email protected])

Last significant revision 4-II-2015