The Prime-perfect Numbers

A Problem Proposal

Consider the numbers n with at least two prime factors, the sum of whose prime factors divides n. In obvious analogy to the perfect numbers, I call these the prime-perfect numbers. (Clearly, the sum of the prime factors of n is almost always less than n, so to require equality of n to the sum, as in the definition of perfect numbers, will be fruitless.)

The sequence a(n) of prime-perfect numbers begins

30, 60, 70, 84, 90, 105, 120, 140, ....

(Note: This is EIS Sequence A066031.) The numbers k with just one prime factor have been excluded from the sequence since they trivially satisfy the requirement that the sum of the prime factors of k divide k. The exclusion thus highlights the interesting numbers satisfying the requirement.

It is easy to see that if p is a prime factor of the prime-perfect number n, then pmn is also prime-perfect for any m. Hence, a is an infinite sequence. But what about the elementary (or primitive) terms of a, that is, terms which are not multiples of any previous terms? For example, 84 is elementary, since it is not a multiple of the preceding terms, 30, 60, 70. But 90 is not elementary because 90 is a multiple of 30. Are there also infinitely many elementary terms?

A related problem: Find an expression generating elementary prime-perfect numbers.

I invite readers to communicate their solutions or comments by contacting me at the email address below. I will report any progress on these problems in this web page, and of course, acknowledge correct solutions.

Joseph L. Pe
iDEN System Engineering Tools and Statistics
Motorola Center
Schaumburg, IL

©2001 J. L. Pe. Document created on 12 December 2001 by J. L. Pe. Last updated on 13 December 2001 2001.

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