12→(1+2+3+4+6)=16→(1+2+4+8)=15→(1+3+5)=9
9→(1+3)=4→(1+2)=3→1.
The trouble is, once you hit a prime, you are finished. The perfect numbers are of course exceptions, each giving us a little loop,while an amicable pair leads to a two-cycle:220→284→220→…. Numbers that lead to cycles longer than two are called sociable. They were not studied at all until the20th century as no one had ever found any. Even today, no number that leads to a three-cycle has been discovered, although there are now 120 known cycles of length four. The first examples were found by P. Poulet in 1918. The first is a five-cycle: