Rules for Constructing Hyperperfect Numbers

摘要

Two rules are given by which hyperperfect numbers (HP's) with k+2 different prime factors can be constructed from certain related numbers with k+1 and with k different prime factors, respectively. By means of these rules many HP's with three and with four, and one with five different prime factors were constructively computed. It is proved that all HP's (below a given bound) can be found with one of these two rules or with an additional rule for the construction of certain HP's. Results are presented of an exhaustive search for all HP's or = 100 million. It is concluded that all the HP's found could also have been computed (but using much less computer time) with at least one of the given rules. A generalization of HP's to hypercycles is described.