almost perfect number

{{Short description|Numbers whose sum of divisors is twice the number minus 1}}

File:Deficient number Cuisenaire rods 8.png, that the number 8 is almost perfect, and deficient.]]

In mathematics, an almost perfect number (sometimes also called slightly defective or least deficient number) is a natural number n such that the sum of all divisors of n (the sum-of-divisors function σ(n)) is equal to 2n − 1, the sum of all proper divisors of n, s(n) = σ(n) − n, then being equal to n − 1. The only known almost perfect numbers are powers of 2 with non-negative exponents {{OEIS|A000079}}. Therefore the only known odd almost perfect number is 2⁰ = 1, and the only known even almost perfect numbers are those of the form 2^k for some positive integer k; 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 six prime factors.{{ 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 | 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 | last=Kishore | first=Masao | title=On odd perfect, quasiperfect, and odd almost perfect numbers | journal=Mathematics of Computation | volume=36 | pages=583–586 | year=1981 | issue=154 | issn=0025-5718 | zbl=0472.10007 | doi=10.2307/2007662| jstor=2007662 | doi-access=free }}

If m is an odd almost perfect number then {{nowrap|m(2m − 1)}} is a Descartes number.{{cite book | last1=Banks | first1=William D. | last2=Güloğlu | first2=Ahmet M. | last3=Nevans | first3=C. Wesley | last4=Saidak | first4=Filip | chapter=Descartes numbers | pages=167–173 | editor1-last=De Koninck | editor1-first=Jean-Marie | editor1-link=Jean-Marie De Koninck | editor2-last=Granville | editor2-first=Andrew | editor2-link=Andrew Granville | editor3-last=Luca | editor3-first=Florian | title=Anatomy of integers. Based on the CRM workshop, Montreal, Canada, March 13–17, 2006 | location=Providence, RI | publisher=American Mathematical Society | series=CRM Proceedings and Lecture Notes | volume=46 | year=2008 | isbn=978-0-8218-4406-9 | zbl=1186.11004 }} Moreover if a and b are positive odd integers such that $b+3 and such that {{nowrap|4$ m − a}} and {{nowrap|4m + b}} are both primes, then {{nowrap|m(4m − a)(4m + b)}} would be an odd weird number.