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Difference between revisions of "Almost perfect number"

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==References==
 
==References==
* Kishore, Masao. "Odd integers N with five distinct prime factors for which $2−10^{−12} < \sigma(N)/N < 2+10^{−12}$". Mathematics of Computation '''32''' (1978) 303–309. ISSN 0025-5718. {{DOI|10.2307/2006281}} {{MR|0485658}}. {{ZBL|0376.10005}}  
+
* Kishore, Masao. "Odd integers N with five distinct prime factors for which $2−10^{−12} < \sigma(N)/N < 2+10^{−12}$". Mathematics of Computation '''32''' (1978) 303–309. {{DOI|10.2307/2006281}} {{MR|0485658}}. {{ZBL|0376.10005}}  
* Kishore, Masao. "On odd perfect, quasiperfect, and odd almost perfect numbers". Mathematics of Computation '''36''' (1981) 583–586. ISSN 0025-5718.  {{ZBL|0472.10007}}  
+
* Kishore, Masao. "On odd perfect, quasiperfect, and odd almost perfect numbers". Mathematics of Computation '''36''' (1981) 583–586. {{ZBL|0472.10007}}  
* Banks, William D.; Güloğlu, Ahmet M.; Nevans, C. Wesley; Saidak, Filip. "Descartes numbers". In De Koninck, Jean-Marie; Granville, Andrew; Luca, Florian (edd), ''Anatomy of integers. Based on the CRM workshop, Montreal, Canada, March 13--17, 2006''. CRM Proceedings and Lecture Notes '''46'''. Providence, RI: American Mathematical Society (2008). pp. 167–173. ISBN 978-0-8218-4406-9{{ZBL|1186.11004}}  
+
* Banks, William D.; Güloğlu, Ahmet M.; Nevans, C. Wesley; Saidak, Filip. "Descartes numbers". In De Koninck, Jean-Marie; Granville, Andrew; Luca, Florian (edd), ''Anatomy of integers. Based on the CRM workshop, Montreal, Canada, March 13--17, 2006''. CRM Proceedings and Lecture Notes '''46'''. Providence, RI: American Mathematical Society (2008). pp. 167–173. {{ISBN|978-0-8218-4406-9}} {{ZBL|1186.11004}}  
 
* Guy, R. K. . "Almost Perfect, Quasi-Perfect, Pseudoperfect, Harmonic, Weird, Multiperfect and Hyperperfect Numbers". Unsolved Problems in Number Theory (2nd ed.). New York: Springer-Verlag (1994). pp. 16, 45–53  
 
* Guy, R. K. . "Almost Perfect, Quasi-Perfect, Pseudoperfect, Harmonic, Weird, Multiperfect and Hyperperfect Numbers". Unsolved Problems in Number Theory (2nd ed.). New York: Springer-Verlag (1994). pp. 16, 45–53  
* Sándor, József; Mitrinović, Dragoslav S.; Crstici, Borislav, edd. (2006). Handbook of number theory I. Dordrecht: Springer-Verlag (2006). p.110. ISBN 1-4020-4215-9{{ZBL|1151.11300}}  
+
* Sándor, József; Mitrinović, Dragoslav S.; Crstici, Borislav, edd. (2006). Handbook of number theory I. Dordrecht: Springer-Verlag (2006). p.110. {{ISBN|1-4020-4215-9}} {{ZBL|1151.11300}}  
* Sándor, Jozsef; Crstici, Borislav, edd. Handbook of number theory II. Dordrecht: Kluwer Academic (2004). pp.37–38. ISBN 1-4020-2546-7. {{ZBL|1079.11001}}
+
* Sándor, Jozsef; Crstici, Borislav, edd. Handbook of number theory II. Dordrecht: Kluwer Academic (2004). pp.37–38. {{ISBN|1-4020-2546-7}} {{ZBL|1079.11001}}

Latest revision as of 19:44, 17 November 2023

2020 Mathematics Subject Classification: Primary: 11A [MSN][ZBL]

Slightly defective number or least deficient number

A natural number $n$ such that the sum of all divisors of $n$ (the sum of divisors function $\sigma(n)$) is equal to $2n − 1$. The only known almost perfect numbers are the powers of 2 with non-negative exponents; however, it has not been shown that all almost perfect numbers are of this form. It is known that an odd almost perfect number greater than 1 would have at least 6 prime factors.

If $m$ is an odd almost perfect number then $m(2m-1)$ is a Descartes number.

References

  • Kishore, Masao. "Odd integers N with five distinct prime factors for which $2−10^{−12} < \sigma(N)/N < 2+10^{−12}$". Mathematics of Computation 32 (1978) 303–309. DOI 10.2307/2006281 MR0485658. Zbl 0376.10005
  • Kishore, Masao. "On odd perfect, quasiperfect, and odd almost perfect numbers". Mathematics of Computation 36 (1981) 583–586. Zbl 0472.10007
  • Banks, William D.; Güloğlu, Ahmet M.; Nevans, C. Wesley; Saidak, Filip. "Descartes numbers". In De Koninck, Jean-Marie; Granville, Andrew; Luca, Florian (edd), Anatomy of integers. Based on the CRM workshop, Montreal, Canada, March 13--17, 2006. CRM Proceedings and Lecture Notes 46. Providence, RI: American Mathematical Society (2008). pp. 167–173. ISBN 978-0-8218-4406-9 Zbl 1186.11004
  • Guy, R. K. . "Almost Perfect, Quasi-Perfect, Pseudoperfect, Harmonic, Weird, Multiperfect and Hyperperfect Numbers". Unsolved Problems in Number Theory (2nd ed.). New York: Springer-Verlag (1994). pp. 16, 45–53
  • Sándor, József; Mitrinović, Dragoslav S.; Crstici, Borislav, edd. (2006). Handbook of number theory I. Dordrecht: Springer-Verlag (2006). p.110. ISBN 1-4020-4215-9 Zbl 1151.11300
  • Sándor, Jozsef; Crstici, Borislav, edd. Handbook of number theory II. Dordrecht: Kluwer Academic (2004). pp.37–38. ISBN 1-4020-2546-7 Zbl 1079.11001
How to Cite This Entry:
Almost perfect number. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Almost_perfect_number&oldid=54523