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, ... (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 one exception, 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 link

• MathWorld: Semiperfect number

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