Semiperfect number

From Wikipedia, the free encyclopedia

Divisibility-based
sets of integers
Form of factorization:
Prime number
Composite number
Powerful number
Square-free number
Achilles number
Constrained divisor sums:
Perfect number
Almost perfect number
Quasiperfect number
Multiply perfect number
Hyperperfect number
Superperfect number
Unitary perfect number
Semiperfect number
Primitive semiperfect number
Practical number
Numbers with many divisors:
Abundant number
Highly abundant number
Superabundant number
Colossally abundant number
Highly composite number
Superior highly composite number
Other:
Deficient number
Weird number
Amicable number
Friendly number
Sociable number
Solitary number
Sublime number
Harmonic divisor number
Frugal number
Equidigital number
Extravagant number
See also:
Divisor function
Divisor
Prime factor
Factorization
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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, ... (sequence A005835 in OEIS);

every multiple of a semiperfect number is semiperfect, and every number of the form 2mp for a natural number m and a prime number p such that p < 2m + 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.

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