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 2^mp 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

• Friedman, Charles N. (1993). "Sums of divisors and Egyptian fractions". Journal of Number Theory 44: 328–339. doi:10.1006/jnth.1993.1057. MR1233293. 

External links

• MathWorld: Semiperfect number