Almost perfect number

A visual example to show that 8 is almost perfect and deficient.

In math, an almost perfect number (also called slightly defective or least deficient number) is a type of natural number n. The sum of n's divisors must be equal to 2n − 1. Every known almost perfect number is a power of 2 and has non-negative exponents (sequence A000079 in the OEIS).

Examples

For example, the divisors of 32 are 1, 2, 4, 8, 16 and 32. The sum of those is 63. 32 ⋅ 2 - 1 is 63. This makes 32 an almost perfect number.

Odd numbers

The only known odd almost perfect number 1. An odd almost perfect number that is not 1 is possible. It would, however, have to have six prime factors.

References

  1. ↑ Kishore, Masao (1978). "Odd integers N with five distinct prime factors for which 2−10−12 < σ(N)/N < 2+10−12" (PDF). Mathematics of Computation. 32: 303–309. doi:10.2307/2006281. ISSN 0025-5718. JSTOR 2006281. MR 0485658. Zbl 0376.10005.
  2. ↑ Kishore, Masao (1981). "On odd perfect, quasiperfect, and odd almost perfect numbers". Mathematics of Computation. 36 (154): 583–586. doi:10.2307/2007662. ISSN 0025-5718. JSTOR 2007662. Zbl 0472.10007.

Further reading