Deficient number

A natural number is called deficient if its real divisor sum (the sum of all divisors without the number itself) is smaller than the number itself. If, on the other hand, the divisor sum is equal to the number, one speaks of a perfect number ; if it is greater, one speaks of an abundant number .

The difference between the real partial sum and the number itself is called deficiency .

Examples

The number 10 is deficient because . It has a deficiency of . ${\displaystyle 1+2+5=8<10}$ ${\displaystyle 10-8=2}$

If the divisor sum is only one less than the number, one speaks of a slightly deficient number (and a deficiency of 1).

All powers of the number 2 are slightly deficient:

power	Divisional sum	Deficiency
${\displaystyle 2^{2}=4}$	${\displaystyle 1+2=3}$	1
${\displaystyle 2^{3}=8}$	${\displaystyle 1+2+4=7}$	1
${\displaystyle 2^{4}=16}$	${\displaystyle 1+2+4+8=15}$	1
${\displaystyle 2^{n}}$	${\displaystyle 2^{n}-1}$	1

The first deficient numbers up to 40 are:

number Divisional sum Deficiency ${\displaystyle 1}$ ${\displaystyle 0}$ ${\displaystyle 1}$ ${\displaystyle 2}$ ${\displaystyle 1}$ ${\displaystyle 1}$ ${\displaystyle 3}$ ${\displaystyle 1}$ ${\displaystyle 2}$ ${\displaystyle 4}$ ${\displaystyle 1+2=3}$ ${\displaystyle 1}$ ${\displaystyle 5}$ ${\displaystyle 1}$ ${\displaystyle 4}$ ${\displaystyle 7}$ ${\displaystyle 1}$ ${\displaystyle 6}$ ${\displaystyle 8}$ ${\displaystyle 1+2+4=7}$ ${\displaystyle 1}$ ${\displaystyle 9}$ ${\displaystyle 1+3=4}$ ${\displaystyle 5}$ ${\displaystyle 10}$ ${\displaystyle 1+2+5=8}$ ${\displaystyle 2}$

number Divisional sum Deficiency ${\displaystyle 11}$ ${\displaystyle 1}$ ${\displaystyle 10}$ ${\displaystyle 13}$ ${\displaystyle 1}$ ${\displaystyle 12}$ ${\displaystyle 14}$ ${\displaystyle 1+2+7=10}$ ${\displaystyle 4}$ ${\displaystyle 15}$ ${\displaystyle 1+3+5=9}$ ${\displaystyle 6}$ ${\displaystyle 16}$ ${\displaystyle 1+2+4+8=15}$ ${\displaystyle 1}$ ${\displaystyle 17}$ ${\displaystyle 1}$ ${\displaystyle 16}$ ${\displaystyle 19}$ ${\displaystyle 1}$ ${\displaystyle 18}$

number Divisional sum Deficiency ${\displaystyle 21}$ ${\displaystyle 1+3+7=11}$ ${\displaystyle 10}$ ${\displaystyle 22}$ ${\displaystyle 1+2+11=14}$ ${\displaystyle 8}$ ${\displaystyle 23}$ ${\displaystyle 1}$ ${\displaystyle 22}$ ${\displaystyle 25}$ ${\displaystyle 1+5=6}$ ${\displaystyle 19}$ ${\displaystyle 26}$ ${\displaystyle 1+2+13=16}$ ${\displaystyle 10}$ ${\displaystyle 27}$ ${\displaystyle 1+3+9=13}$ ${\displaystyle 14}$ ${\displaystyle 29}$ ${\displaystyle 1}$ ${\displaystyle 28}$

number Divisional sum Deficiency ${\displaystyle 31}$ ${\displaystyle 1}$ ${\displaystyle 30}$ ${\displaystyle 32}$ ${\displaystyle 1+2+4+8+16=31}$ ${\displaystyle 1}$ ${\displaystyle 33}$ ${\displaystyle 1+3+11=15}$ ${\displaystyle 18}$ ${\displaystyle 34}$ ${\displaystyle 1+2+17=20}$ ${\displaystyle 14}$ ${\displaystyle 35}$ ${\displaystyle 1+5+7=13}$ ${\displaystyle 22}$ ${\displaystyle 37}$ ${\displaystyle 1}$ ${\displaystyle 36}$ ${\displaystyle 38}$ ${\displaystyle 1+2+19=22}$ ${\displaystyle 16}$ ${\displaystyle 39}$ ${\displaystyle 1+3+13=17}$ ${\displaystyle 22}$

The first deficient numbers are:

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 25, 26, 27, 29, 31, 32, 33, 34, 35, 37, 38, 39, 41, 43, 44, 45, 46, 47, ... sequence A005100 in OEIS

properties

• All prime numbers are deficient because their real divisional sum is always 1.
• There are infinitely many even deficient numbers.
• There are infinitely many odd deficient numbers.
• All odd numbers with one or two different prime factors are deficient numbers.
• All real factors of a deficient number or a perfect number are deficient numbers.
• There is at least one deficient number in the interval for all sufficiently large . ${\displaystyle [n,n+\log(n)^{2}]}$ ${\displaystyle n\geq n_{0}}$

literature

• József Sándor , Dragoslav Mitrinović , Borislav Crstici: Handbook of Number Theory I . 2nd Edition. Springer-Verlag, Dordrecht 2006, ISBN 1-4020-4215-9 , pp. 108 .

Web links

• Deficient number . The Prime Glossary
• Eric W. Weisstein : Deficient Number . In: MathWorld (English).

Individual evidence

1. ↑ József Sándor , Dragoslav Mitrinović , Borislav Crstici: Handbook of Number Theory I. (PDF) (No longer available online.) Springer-Verlag, p. 108 , formerly in the original ; accessed on May 21, 2018 (English).  ( Page no longer available , search in web archives )  Info: The link was automatically marked as defective. Please check the link according to the instructions and then remove this notice.