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 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]

Numbers do exist where the sum of all the divisors σ(n) is equal to 2n + 2: 20, 104, 464, 650, 1952, 130304, 522752 ... (sequence A088831 in the OEIS). Many of these numbers are of the form 2n−1(2n − 3) where 2n − 3 is prime (instead of 2n − 1 with perfect numbers). In addition, numbers exist where the sum of all the divisors σ(n) is equal to 2n - 1, such as the powers of 2.

Notes

  1. ↑ Hagis, Peter; Cohen, Graeme L. (1982). "Some results concerning quasiperfect numbers". J. Austral. Math. Soc. Ser. A. 33 (2): 275–286. doi:10.1017/S1446788700018401. MR 0668448.

References

  • Brown, E.; Abbott, H.; Aull, C.; Suryanarayana, D. (1973). "Quasiperfect numbers" (PDF). Acta Arith. 22: 439–447. doi:10.4064/aa-22-4-439-447. MR 0316368.
  • 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. MR 0485658. Zbl 0376.10005.
  • Cohen, Graeme L. (1980). "On odd perfect numbers (ii), multiperfect numbers and quasiperfect numbers". J. Austral. Math. Soc., Ser. A. 29: 369–384. doi:10.1017/S1446788700021376. ISSN 0263-6115. MR 0569525. Zbl 0425.10005.
  • James J. Tattersall (1999). Elementary number theory in nine chapters. Cambridge University Press. p. 147. ISBN 0-521-58531-7. Zbl 0958.11001.
  • Guy, Richard (2004). Unsolved Problems in Number Theory, third edition. Springer-Verlag. p. 74. ISBN 0-387-20860-7.
  • Sándor, József; Mitrinović, Dragoslav S.; Crstici, Borislav, eds. (2006). Handbook of number theory I. Dordrecht: Springer-Verlag. pp. 109–110. ISBN 1-4020-4215-9. Zbl 1151.11300.


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