sentences:  A natural number formula_2 is a sociable factorion if it is a periodic point for formula_14, where formula_26 for a positive integer formula_23, and forms a cycle of period formula_23.A factorion is a sociable factorion with formula_29, and a amicable factorion is a sociable factorion with formula_30.All natural numbers formula_2 are preperiodic points for formula_14, regardless of the base.