Quasiperfect number explained

In mathematics, a quasiperfect number is a natural number n for which the sum of all its divisors (the sum-of-divisors function σ(n)) is equal to 2n + 1. Equivalently, n is the sum of its non-trivial divisors (that is, its divisors excluding 1 and n). No quasiperfect numbers have been found so far.

The quasiperfect numbers are the abundant numbers of minimal abundance (which is 1).

Theorems

If a quasiperfect number exists, it must be an odd square number greater than 10³⁵ and have at least seven distinct prime factors.^[1]

Related

For a perfect number n the sum of all its divisors is equal to 2n.

For an almost perfect number n the sum of all its divisors is equal to 2n - 1.

Betrothed numbers relate to quasiperfect numbers like amicable numbers relate to perfect numbers.

Notes

1. Hagis. Peter . Cohen. Graeme L.. Some results concerning quasiperfect numbers. J. Austral. Math. Soc. Ser. A. 33. 1982. 275–286. 10.1017/S1446788700018401. 2. 0668448. free.

References

• E. . Brown. H. . Abbott. C. . Aull. D. . Suryanarayana. Quasiperfect numbers. Acta Arith.. 1973. 22. 4. 439–447. 0316368. 10.4064/aa-22-4-439-447. free.
• Kishore . Masao . Odd integers N with five distinct prime factors for which 2−10⁻¹² < σ(N)/N < 2+10⁻¹² . . 32 . 141 . 303–309 . 1978 . 0025-5718 . 0376.10005 . 0485658. 10.2307/2006281. 2006281 .
• Graeme L.. Cohen. On odd perfect numbers (ii), multiperfect numbers and quasiperfect numbers . 1980. J. Austral. Math. Soc. Ser. A . 29 . 3. 369–384 . 10.1017/S1446788700021376 . 0569525 . 0425.10005 . 120459203. 0263-6115 .
• Book: James J. Tattersall . Elementary number theory in nine chapters . limited. . 0-521-58531-7 . 1999 . 147. 0958.11001 .
• Book: Guy , Richard . Richard K. Guy . 2004 . Unsolved Problems in Number Theory, third edition . 74 . . 0-387-20860-7.
• Book: Sándor . József . Mitrinović . Dragoslav S. . Crstici . Borislav . Handbook of number theory I . Dordrecht . . 2006 . 1-4020-4215-9 . 1151.11300 . 109–110.