Abundant number

A natural number is abundant ( lat. Abundans "overloaded") when their real divider sum (the sum of all the divisors excluding the number itself) is greater than the number itself. If the divider total contrast, equal to the number, it is called a perfect number , if it is smaller, one speaks of a deficient number .

A number n is called slightly abundant or it is called a quasi-perfect number if the sum of its real divisors equals n +1. The question of whether there is a slightly abundant number has not yet been clarified. It would have to be an odd square number that is greater than and has at least seven different prime factors. ${\displaystyle 10^{35}}$

An abundant number that is not a pseudo-perfect number ( i.e. cannot be represented as the sum of several different real divisors) is called a strange number .

The difference between the true divisional sum and the number itself is called abundance .

Examples

The number 20 is abundant because 1 + 2 + 4 + 5 + 10 = 22> 20. It has an abundance of 22-20 = 2.

The first abundant numbers up to 100 are:

number real partial sum Abundance ${\displaystyle 12}$ ${\displaystyle 1+2+3+4+6=16}$ ${\displaystyle 4}$ ${\displaystyle 18}$ ${\displaystyle 1+2+3+6+9=21}$ ${\displaystyle 3}$ ${\displaystyle 20}$ ${\displaystyle 1+2+4+5+10=22}$ ${\displaystyle 2}$ ${\displaystyle 24}$ ${\displaystyle 1+2+3+4+6+8+12=36}$ ${\displaystyle 12}$ ${\displaystyle 30}$ ${\displaystyle 1+2+3+5+6+10+15=42}$ ${\displaystyle 12}$ ${\displaystyle 36}$ ${\displaystyle 1+2+3+4+6+9+12+18=55}$ ${\displaystyle 19}$ ${\displaystyle 40}$ ${\displaystyle 1+2+4+5+8+10+20=50}$ ${\displaystyle 10}$ ${\displaystyle 42}$ ${\displaystyle 1+2+3+6+7+14+21=54}$ ${\displaystyle 12}$ ${\displaystyle 48}$ ${\displaystyle 1+2+3+4+6+8+12+16+24=76}$ ${\displaystyle 28}$ ${\displaystyle 54}$ ${\displaystyle 1+2+3+6+9+18+27=66}$ ${\displaystyle 12}$ ${\displaystyle 56}$ ${\displaystyle 1+2+4+7+8+14+28=64}$ ${\displaystyle 8}$

number real partial sum Abundance ${\displaystyle 60}$ ${\displaystyle 1+2+3+4+5+6+10+12+15+20+30=108}$ ${\displaystyle 48}$ ${\displaystyle 66}$ ${\displaystyle 1+2+3+6+11+22+33=78}$ ${\displaystyle 12}$ ${\displaystyle 70}$ ${\displaystyle 1+2+5+7+10+14+35=74}$ ${\displaystyle 4}$ ${\displaystyle 72}$ ${\displaystyle 1+2+3+4+6+8+9+12+18+24+36=123}$ ${\displaystyle 51}$ ${\displaystyle 78}$ ${\displaystyle 1+2+3+6+13+26+39=90}$ ${\displaystyle 12}$ ${\displaystyle 80}$ ${\displaystyle 1+2+4+5+8+10+16+20+40=106}$ ${\displaystyle 26}$ ${\displaystyle 84}$ ${\displaystyle 1+2+3+4+6+7+12+14+21+28+42=140}$ ${\displaystyle 56}$ ${\displaystyle 88}$ ${\displaystyle 1+2+4+8+11+22+44=92}$ ${\displaystyle 4}$ ${\displaystyle 90}$ ${\displaystyle 1+2+3+5+6+9+10+15+18+30+45=144}$ ${\displaystyle 54}$ ${\displaystyle 96}$ ${\displaystyle 1+2+3+4+6+8+12+16+24+32+48=156}$ ${\displaystyle 60}$ ${\displaystyle 100}$ ${\displaystyle 1+2+4+5+10+20+25+50=117}$ ${\displaystyle 17}$

The first abundant numbers are:

12, 18, 20, 24, 30, 36, 40, 42, 48, 54, 56, 60, 66, 70, 72, 78, 80, 84, 88, 90, 96, 100, 102, 104, 108, 112, 114, 120, 126, 132, 138, ... sequence A005101 in OEIS

The first odd abundant numbers are

945, 1575, 2205, 2835, 3465, 4095, 4725, 5355, 5775, 5985, 6435, 6615, 6825, 7245, 7425, 7875, 8085, 8415, 8505, 8925, 9135, ... (sequence A005231 in OEIS )

The smallest abundant number is 12 (real partial sum 1 + 2 + 3 + 4 + 6 = 16> 12).

The smallest abundant number that cannot be divided by 3 is 20 (real divisor sum 1 + 2 + 4 + 5 + 10 = 22> 20)

The smallest odd abundant number is 945 (real partial sum 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 63 + 105 + 135 + 189 + 315 = 975> 945).

The smallest odd abundant number that is not divisible by 3 is , whose real divisor sum is. ${\displaystyle 5.391.411.025=5^{2}\cdot 7\cdot 11\cdot 13\cdot 17\cdot 19\cdot 23\cdot 29}$ ${\displaystyle 5.407.897.775}$

The following is a list of the smallest abundant numbers, which are not divisible by the first n prime numbers:

12, 945, 5391411025, 20169691981106018776756331, 49061132957714428902152118459264865645885092682687973 ,, ... (Follow A047802 in OEIS )

The smallest abundant number divisible by k is at most 6k (1 + 2 + 3 + 6 + k + 2k + 3k = 6k + 12> 6k)

properties

• There are an infinite number of even abundant numbers.
• There are an infinite number of odd abundant numbers.
• Every multiple (> 1) of a perfect number is abundant. (For example, every multiple of 6 is abundant, because the divisors of these multiples also include the divisors and , which already result as a sum .) ${\displaystyle 1,{\frac {n}{2}},{\frac {n}{3}}}$ ${\displaystyle {\frac {n}{6}}}$ ${\displaystyle 1+{\frac {n}{2}}+{\frac {n}{3}}+{\frac {n}{6}}={\frac {6n+6}{6}}=n+1>n}$
• Any multiple of an abundant number is abundant. (For example, every multiple of 20 is abundant (including the 20 itself) because the divisors of these multiples also include the divisors and , which in themselves result as a sum .) ${\displaystyle {\frac {n}{2}},{\frac {n}{4}},{\frac {n}{5}},{\frac {n}{10}}}$ ${\displaystyle {\frac {n}{20}}}$ ${\displaystyle {\frac {n}{2}}+{\frac {n}{4}}+{\frac {n}{5}}+{\frac {n}{10}}+{\frac {n}{20}}={\frac {22n}{20}}=n(1+{\frac {1}{10}})>n}$
• Every whole number> 20161 can be written as the sum of two abundant numbers. The only 1456 smaller numbers that cannot be written as the sum of two abundant numbers are as follows:

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 31, 33, 34, 35, 37, 39, 41, 43,…, 20161 (episode A048242 in OEIS )

literature

• Douglas E. Iannucci: On the smallest abundant number not divisible by the first k primes . In: Bulletin of the Belgian Mathematical Society . tape 12 , no. 1 , 2005, p. 39-44 .

Web links

• Eric W. Weisstein : Abundant number . In: MathWorld (English).
• Peter Hagis Jr., Graeme L. Cohen: Some results concerning quasiperfect numbers. Journal of the Australian Mathematical Society , pp. 275–286 , accessed May 21, 2018 . 
• Douglas E. Iannucci: On the smallest abundant number not divisible by the first k primes. Bulletin of the Belgian Mathematical Society , pp. 39-44 , accessed on May 21, 2018 . 

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 .