Quasiperfect number

Short description: Numbers whose sum of divisors is twice the number plus 1

Unsolved problem in mathematics: Do quasiperfect numbers exist? (more unsolved problems in mathematics)

In mathematics, a quasiperfect number is a natural number n for which the sum of all its divisors (the sum-of-divisors function $\sigma (n)$) 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.

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 $2n-1$.

Numbers n whose sum of factors equals $2n+2$ are known to exist. They are of form $2^{n-1}\times (2^n-3)$ where $2^n-3$ is a prime. The only exception known so far is $650=2\times 5^2\times 13$. They are 20, 104, 464, 650, 1952, 130304, 522752, ... (sequence A088831 in the OEIS). Numbers n whose sum of factors equals $2n-2$ are also known to exist. They are of form $2^{n-1}\times (2^n+1)$ where $2^n+1$ is prime. No exceptions are found so far. Because of the five known Fermat primes, there are five such numbers known: 3, 10, 136, 32896 and 2147516416 (sequence A191363 in the OEIS)

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

References

• Brown, E.; Abbott, H.; Aull, C.; Suryanarayana, D. (1973). "Quasiperfect numbers". Acta Arith. 22 (4): 439–447. doi:10.4064/aa-22-4-439-447. http://matwbn.icm.edu.pl/ksiazki/aa/aa22/aa2245.pdf. 
• Kishore, Masao (1978). "Odd integers N with five distinct prime factors for which 2−10⁻¹² < σ(N)/N < 2+10⁻¹²". Mathematics of Computation 32 (141): 303–309. doi:10.2307/2006281. ISSN 0025-5718. https://www.ams.org/journals/mcom/1978-32-141/S0025-5718-1978-0485658-X/S0025-5718-1978-0485658-X.pdf. 
• Cohen, Graeme L. (1980). "On odd perfect numbers (ii), multiperfect numbers and quasiperfect numbers". J. Austral. Math. Soc. Ser. A 29 (3): 369–384. doi:10.1017/S1446788700021376. ISSN 0263-6115. 
• James J. Tattersall (1999). Elementary number theory in nine chapters. Cambridge University Press. pp. 147. ISBN 0-521-58531-7. https://archive.org/details/elementarynumber00tatt_470. 
• 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.