A
perfect number is an integer that is exactly equal to the sum of its proper divisors; that is, it is equal to the sum of all those integers that divide it without leaving a remainder, excluding the number itself.
For example, the proper divisors of 6 are 1, 2, and 3; and 6 = 1 + 2 + 3.
For another example, the proper divisors of 28 are 1, 2, 4, 7, and 14; and 28 = 1 + 2 + 4 + 7 + 14.
An integer that exceeds the sum of its proper divisors is called "excessive" (or "abundant"). One example is 12, whose proper divisors are 1, 2, 3, 4, and 6.
An integer that is less than the sum of its proper divisors is called "deficient". One example is 10, whose proper divisors are 1, 2, and 5.
Perfect integers occur much less frequently than excessive or deficient integers. The ones mentioned in my earlier post are the only examples less than 100,000,000.
__________________
Richard Wallace
Mentor since 2011 for FRC 3620 Average Joes (St. Joseph, Michigan)
Mentor 2002-10 for FRC 931 Perpetual Chaos (St. Louis, Missouri)
since 2003
I believe in intuition and inspiration. Imagination is more important than knowledge. For knowledge is limited, whereas imagination embraces the entire world, stimulating progress, giving birth to evolution. It is, strictly speaking, a real factor in scientific research.
(Cosmic Religion : With Other Opinions and Aphorisms (1931) by Albert Einstein, p. 97)