A347770 Conjectured list of numbers which are perfect, amicable, or sociable.

6, 28, 220, 284, 496, 1184, 1210, 2620, 2924, 5020, 5564, 6232, 6368, 8128, 10744, 10856, 12285, 12496, 14264, 14288, 14316, 14536, 14595, 15472, 17296, 17716, 18416, 19116, 19916, 22744, 22976, 31704, 45946, 47616, 48976, 63020, 66928, 66992, 67095, 69615, 71145, 76084, 79750

Offset

1,1

Comments

By definition, this is the union of A000396, A259180, and A122726. However, at present A122726 is not known to be complete. There is no proof that 564 (for example) is missing from this sequence. - N. J. A. Sloane, Sep 17 2021

Numbers m for which there exists k>=1 such that s^k(m) = m, where s is A001065.

Conjecture: There are no aliquot cycles containing even numbers and odd numbers simultaneously, i.e., every aliquot cycle either has only even numbers or has only odd numbers.

Links

Table of n, a(n) for n=1..43.

David Moews, A list of amicable pairs below 2.01 * 10^11

David Moews, A list of the first 5001 amicable pairs

David Moews, A list of currently known aliquot cycles of length greater than 2 [This list is not known to be complete.]

Jan Munch Pedersen, Tables of Aliquot Cycles (from wayback machine)

Eric Weisstein's World of Mathematics, Perfect number

Eric Weisstein's World of Mathematics, Amicable Pair

Eric Weisstein's World of Mathematics, Sociable Numbers

Wikipedia, Perfect number

Wikipedia, Amicable number

Wikipedia, Sociable number

Example

Known aliquot cycles (sorted by smallest member):
{6}
{28}
{220, 284}
{496}
{1184, 1210}
{2620, 2924}
{5020, 5564}
{6232, 6368}
{8128}
{10744, 10856}
{12285, 14595}
{12496, 14288, 15472, 14536, 14264}
{14316, 19116, 31704, 47616, 83328, 177792, 295488, 629072, 589786, 294896, 358336, 418904, 366556, 274924, 275444, 243760, 376736, 381028, 285778, 152990, 122410, 97946, 48976, 45946, 22976, 22744, 19916, 17716}
{17296, 18416}
...

Crossrefs

Cf. A000396, A002025, A002046, A003416, A063990, A122726, A206708, A259180.

Sequence in context: A169723 A052395 A034660 * A206708 A336565 A216413

Adjacent sequences: A347767 A347768 A347769 * A347771 A347772 A347773

Keyword

nonn

Author

Eric Chen, Sep 13 2021

Extensions

Edited with new definition (pointing out that the list is only conjectured to be complete) by N. J. A. Sloane, Sep 17 2021

Status

approved