Quasiperfect number

In mathematics, a quasiperfect number is a natural number n for which the sum of all its divisors (the divisor 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

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.

Numbers do exist where the sum of all the divisors is equal to 2n + 2: 20, 104, 464, 650, 1952, 130304, 522752 ... (sequence A088831 in the OEIS). All numbers of the form 2^{n − 1}(2ⁿ − 3) where 2ⁿ − 3 is prime belong to the sequence. As of 2024^[update], the only known number of a different form in the sequence is 650 = 2 * 5² * 13.

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

Notes

References

• Brown, E.; Abbott, H.; Aull, C.; Suryanarayana, D. (1973). "Quasiperfect numbers" (PDF). Acta Arith. 22 (4): 439–447.
  MR 0316368
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• Kishore, Masao (1978). "Odd integers N with five distinct prime factors for which 2−10⁻¹² < σ(N)/N < 2+10⁻¹²" (PDF).
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• Cohen, Graeme L. (1980). "On odd perfect numbers (ii), multiperfect numbers and quasiperfect numbers". J. Austral. Math. Soc. Ser. A. 29 (3): 369–384.
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• James J. Tattersall (1999). Elementary number theory in nine chapters.
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• ISBN 0-387-20860-7
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• Sándor, József; Mitrinović, Dragoslav S.; Crstici, Borislav, eds. (2006). Handbook of number theory I. Dordrecht:
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