Almost perfect number

In mathematics, an almost perfect number (sometimes also called slightly defective or least deficient number) is a natural number n such that the sum of all divisors of n (the sum-of-divisors function σ(n)) is equal to 2n − 1, the sum of all proper divisors of n, s(n) = σ(n) − n, then being equal to n − 1. The only known almost perfect numbers are powers of 2 with non-negative exponents (sequence A000079 in the OEIS). Therefore the only known odd almost perfect number is 2⁰ = 1, and the only known even almost perfect numbers are those of the form 2^k for some positive integer k; however, it has not been shown that all almost perfect numbers are of this form. It is known that an odd almost perfect number greater than 1 would have at least six prime factors.^[1][2]

If m is an odd almost perfect number then m(2m − 1) is a Descartes number.^[3] Moreover if a and b are positive odd integers such that ${\displaystyle b+3<a<{\sqrt {m/2}}}$ and such that 4m − a and 4m + b are both primes, then m(4m − a)(4m + b) would be an odd weird number.^[4]

See also

• Perfect number
• Quasiperfect number

References

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.

External links

• Weisstein, Eric W. "Almost perfect number". MathWorld.

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This page was last edited on 2 March 2023, at 07:05