Quasi-perfect number

In mathematics , natural numbers n are called quasi-perfect numbers or quasi-perfect numbers if the sum of their real divisors (i.e. all divisors except for the number n itself) results in n + 1 (i.e. if the divisor-sum function or if is). There are still no quasi-perfect numbers known. ${\displaystyle \sigma (n)=2n+1}$ ${\displaystyle \sigma ^{*}(n)=n+1}$

properties

• Quasi-perfect numbers are abundant numbers with an abundance of 1. That is why they are also called slightly abundant numbers .
• Quasi-perfect numbers must be odd square numbers that are greater than and have at least seven different prime factors . ${\displaystyle 10^{35}}$

Similarity to other numbers

There are numbers that have an abundance of 2, the real sum of which is n + 2 . The first of these numbers are the following:

20, 104, 464, 650, 1952, 130304, 522752, 8382464, 134193152, 549754241024, 8796086730752, 140737463189504, ... sequence A088831 in OEIS

literature

• József Sándor , Dragoslav Mitrinović , Borislav Crstici: Handbook of Number Theory I . 2nd Edition. Springer-Verlag, Dordrecht 2006, ISBN 1-4020-4215-9 , pp. 109-110 .
• Masao Kishore: Odd integers N with Five Distinct Prime Factors for Which 2 −10 ⁻¹² <σ ( N ) / N <2 + 10 ⁻¹² . In: Mathematics of Computation . tape 32 , 1978, p. 303-309 .
• HL Abbott, CE Aull, Ezra Brown, D. Suryanarayana: Quasiperfect numbers . In: Acta Arithmetica . tape 22 , 1973, p. 439-447 .

Web links

• Peter Hagis Jr., Graeme L. Cohen: Some results concerning quasiperfect numbers. Journal of the Australian Mathematical Society , pp. 275–286 , accessed May 21, 2018 . 
• József Sándor , Dragoslav Mitrinović , Borislav Crstici: Handbook of Number Theory I. (PDF) Springer-Verlag, pp. 109–110 , accessed on May 21, 2018 (English). 
• Masao Kishore: Odd integers N with Five Distinct Prime Factors for Which 2−10 ⁻¹² <σ ( N ) / N <2 + 10 ⁻¹² . (PDF) Mathematics of Computation, pp. 303–309 , accessed on May 24, 2018 (English). 
• HL Abbott, CE Aull, Ezra Brown, D. Suryanarayana: Quasiperfect numbers. (PDF) Acta Arithmetica , pp. 439–447 , accessed on May 24, 2018 (English). 
• Eric W. Weisstein : Quasiperfect Number . In: MathWorld (English).
• Quasiperfect Number . In: PlanetMath . (English)

Individual evidence

1. ↑ Peter Hagis Jr., Graeme L. Cohen: Some results concerning quasiperfect numbers . In: Journal of the Australian Mathematical Society . tape 33 , no. 2 , 1982, p. 275-286 .