Semiperfect number

Semiperfect number

In mathematics, a semiperfect number or pseudoperfect number is a natural number "n" that is equal to the sum of all or some of its proper divisors.

The first few semiperfect numbers are:6, 12, 18, 20, 24, 28, 30, 36, 40, ... OEIS|id=A005835;every multiple of a semiperfect number is semiperfect, and every number of the form 2"m""p" for a natural number "m" and a prime number "p" such that "p" < 2"m" + 1 is also semiperfect.

The smallest odd semiperfect number is 945 (see, e.g., Friedman 1993).

A semiperfect number that is equal to the sum of all its proper divisors is called a perfect number; an abundant number which is not semiperfect is called a weird number. With the exception of 2, all primary pseudoperfect numbers are semiperfect. Every practical number that is not a power of two is semiperfect.

A semiperfect number that is not divisible by any smaller semiperfect number is a primitive semiperfect number.

References

* cite journal
title = Sums of divisors and Egyptian fractions
author = Friedman, Charles N.
journal = Journal of Number Theory
year = 1993
volume = 44
pages = 328–339
url = http://dell5.ma.utexas.edu/users/friedman/divisors.ps
id = MathSciNet | id = 1233293
doi = 10.1006/jnth.1993.1057

External links

* [http://mathworld.wolfram.com/SemiperfectNumber.html MathWorld: Semiperfect number]


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