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

Author

Eric Chen, Sep 13 2021

Keywords

• nonn

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.

Examples

			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}
...
		

Links

• 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

Crossrefs

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

Extensions

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