Tables of Aliquot Cycles

Click on the number of known cycles to view the corresponding lists.

Type Amicable Numbers Sociable Numbers
of order four
Sociable Numbers
of other orders
Perfect Numbers
Known
Cycles
Last
Update
Known
Cycles
Last
Update
Known
Cycles
Last
Update
Known
Cycles
Last
Update
Ordinary 11994387  28-Sep-2007 142  01-Oct-2007 10  13-Nov-2006 44  11-Sep-2006
Unitary 4911908  28-Sep-2007 191  20-Nov-2006 24  14-Jun-2005 5  14-Dec-1997
Infinitary 11538100  28-Sep-2007 5034  28-Sep-2007 129  20-Nov-2006 190  28-Sep-2007
Exponential (note 2) 3089296  28-Sep-2007 371  20-Nov-2006 38  15-Jun-2005 12  06-Dec-1998
Augmented 1931  05-Feb-2002 2  08-May-2003 0  18-Oct-1997 note 1  11-Mar-1998
Augmented Unitary 27  05-Feb-2002 0  06-Dec-1998 0  06-Dec-1998
Augmented Infinitary 425  10-Sep-2003 0  06-Dec-1998 0  06-Dec-1998
Reduced 1946  15-Feb-2003 0  18-Oct-1997 1  18-Oct-1997 0  11-Mar-1998
Reduced Unitary 28  15-Feb-2003 0  06-Dec-1998 0  06-Dec-1998
Reduced Infinitary 427  10-Sep-2003 0  06-Dec-1998 0  06-Dec-1998

Note 1: All powers of 2 are augmented perfect numbers; no other augmented perfect numbers are known.

Note 2: Multiplying all members of an exponential aliquot cycle by a squarefree number prime to all the members of the cycle yields a new exponential aliquot cycle. Cycles that can be constructed from other cycles this way aren't listed (nor counted).

Some of the discoverer information in the lists might still be wrong. All kind of updates are welcome. A great thanks to David Moews for sending me a lot of discoverer information.


Exhaustive limits

This table indicates how far exhaustive searchs for the various kinds of cycles has been carried. For amicable numbers all pairs with smaller member below the limit are known. For sociable numbers all cycles with the member preceeding the largest member below the limit are known.

Type Amicable Numbers
Exhaustive below
Sociable Numbers
Exhaustive below
Perfect Numbers
Exhaustive below
Ordinary 1014 5×1012 10300
Unitary 2×1012 2×1011 (2×1012)
Infinitary 5×1012 2×1011 (5×1012)
Exponential (4×1011) (4×1011) (4×1011)
Augmented 1012 2×1011 (1012)
Augmented Unitary 2×1011 2×1011
Augmented Infinitary 2×1011 2×1011
Reduced 1012 2×1011 1035
Reduced Unitary 2×1011 2×1011
Reduced Infinitary 2×1011 2×1011

Links


Last update: 01-Oct-2007

Jan Munch Pedersen, amicable@post.cybercity.dk