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's not 1 is possible. It would, however, have to have six prime factors.

Further reading

• Guy, R. K. (1994). "Almost Perfect, Quasi-Perfect, Pseudoperfect, Harmonic, Weird, Multiperfect and Hyperperfect Numbers". Unsolved Problems in Number Theory (2nd ed.). New York: Springer-Verlag. pp. 16, 45–53.
• Sándor, József; Mitrinović, Dragoslav S.; Crstici, Borislav, eds. (2006). Handbook of number theory I. Dordrecht: Springer-Verlag. p. 110. ISBN 1-4020-4215-9. Zbl 1151.11300.
• Sándor, Jozsef; Crstici, Borislav, eds. (2004). Handbook of number theory II. Dordrecht: Kluwer Academic. pp. 37–38. ISBN 1-4020-2546-7. Zbl 1079.11001.
• Singh, S. (1997). Fermat's Enigma: The Epic Quest to Solve the World's Greatest Mathematical Problem. New York: Walker. p. 13. ISBN 9780802713315.