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:Arithmetic dynamics

Category:Divisor function

Category:Integer sequences

Category:Unsolved problems in mathematics

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