Quasiperfect number
{{Short description|Numbers whose sum of divisors is twice the number plus 1}}
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.{{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}}
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 exist whose sum of factors = 2n + 2. They are of form 2^(n - 1) * (2^n - 3) where 2^n - 3 is a prime. The only exception known till yet is 650 = 2 * 5^2 * 13. They are 20, 104, 464, 650, 1952, 130304, 522752, etc. (OEIS A088831) Numbers n exist whose sum of factors = 2n - 2. They are of form 2^(n - 1) * (2^n + 1) where 2^n + 1 is prime. No exceptions are found till yet. Because of 5 known Fermat Primes, there are 5 known such numbers: 3, 10, 136, 32896 and 2147516416. (OEIS A191363)
Betrothed numbers relate to quasiperfect numbers like amicable numbers relate to perfect numbers.
Notes
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 2−10−12 < σ(N)/N < 2+10−12
| 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:Unsolved problems in mathematics
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