{{Short description|Numbers whose sum of divisors is twice the number plus 1}} {{unsolved|mathematics|Do quasiperfect numbers exist?}} In mathematics, a '''quasiperfect number''' is a natural number {{mvar|n}} for which the sum of all its divisors (the sum-of-divisors function <math>\sigma(n)</math>) is equal to <math>2n + 1</math>. Equivalently, {{mvar|n}} is the sum of its non-trivial divisors (that is, its divisors excluding 1 and {{mvar|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 10<sup>35</sup> and have at least seven distinct prime factors.<ref>{{cite journal|last1=Hagis|first1=Peter |last2=Cohen|first2=Graeme L.|title=Some results concerning quasiperfect numbers|journal=J. Austral. Math. Soc. Ser. A|volume=33|year=1982|pages=275–286|doi=10.1017/S1446788700018401|issue=2|mr=0668448|doi-access=free}}</ref>
== Related == For a perfect number {{mvar|n}} the sum of all its divisors is equal to <math>2n</math>. For an almost perfect number {{mvar|n}} the sum of all its divisors is equal to <math>2n - 1</math>.
Numbers {{mvar|n}} whose sum of factors equals <math>2n + 2</math> are known to exist. They are of form <math>2^{n - 1} \times (2^n - 3)</math> where <math>2^n - 3</math> is a prime. The only exception known so far is <math>650 = 2 \times 5^2 \times 13</math>. They are 20, 104, 464, 650, 1952, 130304, 522752, ... {{OEIS|A088831}}. Numbers {{mvar|n}} whose sum of factors equals <math>2n - 2</math> are also known to exist. They are of form <math>2^{n - 1} \times (2^n + 1)</math> where <math>2^n + 1</math> 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 {{OEIS|A191363}}
Betrothed numbers relate to quasiperfect numbers like amicable numbers relate to perfect numbers.
==Notes== <references/>
== References == * {{cite journal|first1=E. |last1=Brown |first2=H. |last2=Abbott |first3=C. |last3=Aull |first4=D. |last4=Suryanarayana |title=Quasiperfect numbers |journal=Acta Arith. |year=1973 |volume=22 |issue=4 |pages=439–447 |mr=0316368 |url=http://matwbn.icm.edu.pl/ksiazki/aa/aa22/aa2245.pdf |doi=10.4064/aa-22-4-439-447 |doi-access=free }} * {{cite journal | last=Kishore | first=Masao | title=Odd integers ''N'' with five distinct prime factors for which 2−10<sup>−12</sup> < σ(''N'')/''N'' < 2+10<sup>−12</sup> | journal=Mathematics of Computation | volume=32 | issue=141 | pages=303–309 | year=1978 | issn=0025-5718 | zbl=0376.10005 | mr=0485658 | url=https://www.ams.org/journals/mcom/1978-32-141/S0025-5718-1978-0485658-X/S0025-5718-1978-0485658-X.pdf | doi=10.2307/2006281 | jstor=2006281 }} * {{cite journal|first1=Graeme L. |last1=Cohen|title= On odd perfect numbers (ii), multiperfect numbers and quasiperfect numbers |year=1980 |journal=J. Austral. Math. Soc. Ser. A |volume=29 |issue=3|pages=369–384 |doi=10.1017/S1446788700021376 | mr=0569525 | zbl=0425.10005 |s2cid=120459203| issn=0263-6115 }} * {{cite book | author=James J. Tattersall | title=Elementary number theory in nine chapters | url=https://archive.org/details/elementarynumber00tatt_470 | url-access=limited | publisher=Cambridge University Press | isbn=0-521-58531-7 | year=1999 | pages=[https://archive.org/details/elementarynumber00tatt_470/page/n156 147] | zbl=0958.11001 }} * {{cite book | last = Guy | first = Richard | author-link = Richard K. Guy | year = 2004 | title = Unsolved Problems in Number Theory, third edition |page=74 | publisher = Springer-Verlag | isbn=0-387-20860-7 }} * {{cite book | editor1-last=Sándor | editor1-first=József | editor2-last=Mitrinović | editor2-first=Dragoslav S. | editor3-last=Crstici |editor3-first=Borislav | title=Handbook of number theory I | location=Dordrecht | publisher=Springer-Verlag | year=2006 | isbn=1-4020-4215-9 | zbl=1151.11300 | pages=109–110 }}
{{Divisor classes}} {{Classes of natural numbers}}
Category:Arithmetic dynamics Category:Divisor function Category:Integer sequences Category:Unsolved problems in number theory