Demonstration, with Cuisenaire rods, of the abundance of the number 12
Demonstration, with Cuisenaire rods, of the abundance of the number 12

In number theory, an abundant number or excessive number is a number for which the sum of its proper divisors is greater than the number. The integer 12 is the first abundant number. Its proper divisors are 1, 2, 3, 4 and 6 for a total of 16. The amount by which the sum exceeds the number is the abundance. The number 12 has an abundance of 4, for example.

Definition

A number n for which the sum of divisors σ(n) > 2n, or, equivalently, the sum of proper divisors (or aliquot sum) s(n) > n.

Abundance is the value σ(n) − 2n (or s(n) − n).

Examples

The first 28 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, ... (sequence A005101 in the OEIS).

For example, the proper divisors of 24 are 1, 2, 3, 4, 6, 8, and 12, whose sum is 36. Because 36 is greater than 24, the number 24 is abundant. Its abundance is 36 − 24 = 12.

Properties

for sufficiently large k.
Let 
  
    
      
        a
        (
        n
        )
      
    
    {\displaystyle a(n)}
  
 be the number of abundant numbers not exceeding 
  
    
      
        n
      
    
    {\displaystyle n}
  
. Plot of 
  
    
      
        a
        (
        n
        )
        
          /
        
        n
      
    
    {\displaystyle a(n)/n}
  
 for 
  
    
      
        n
        <
        
          10
          
            6
          
        
      
    
    {\displaystyle n<10^{6))
  
 (with 
  
    
      
        n
      
    
    {\displaystyle n}
  
  log-scaled)
Let be the number of abundant numbers not exceeding . Plot of for (with log-scaled)

Related concepts

Euler diagram of abundant, primitive abundant, highly abundant, superabundant, colossally abundant, highly composite, superior highly composite, weird and perfect numbers under 100 in relation to deficient and composite numbers

Numbers whose sum of proper factors equals the number itself (such as 6 and 28) are called perfect numbers, while numbers whose sum of proper factors is less than the number itself are called deficient numbers. The first known classification of numbers as deficient, perfect or abundant was by Nicomachus in his Introductio Arithmetica (circa 100 AD), which described abundant numbers as like deformed animals with too many limbs.

The abundancy index of n is the ratio σ(n)/n.[7] Distinct numbers n1, n2, ... (whether abundant or not) with the same abundancy index are called friendly numbers.

The sequence (ak) of least numbers n such that σ(n) > kn, in which a2 = 12 corresponds to the first abundant number, grows very quickly (sequence A134716 in the OEIS).

The smallest odd integer with abundancy index exceeding 3 is 1018976683725 = 33 × 52 × 72 × 11 × 13 × 17 × 19 × 23 × 29.[8]

If p = (p1, ..., pn) is a list of primes, then p is termed abundant if some integer composed only of primes in p is abundant. A necessary and sufficient condition for this is that the product of pi/(pi − 1) be > 2.[9]

References

  1. ^ D. Iannucci (2005), "On the smallest abundant number not divisible by the first k primes", Bulletin of the Belgian Mathematical Society, 12 (1): 39–44
  2. ^ a b Tattersall (2005) p.134
  3. ^ Hall, Richard R.; Tenenbaum, Gérald (1988). Divisors. Cambridge Tracts in Mathematics. Vol. 90. Cambridge: Cambridge University Press. p. 95. ISBN 978-0-521-34056-4. Zbl 0653.10001.
  4. ^ Deléglise, Marc (1998). "Bounds for the density of abundant integers". Experimental Mathematics. 7 (2): 137–143. CiteSeerX 10.1.1.36.8272. doi:10.1080/10586458.1998.10504363. ISSN 1058-6458. MR 1677091. Zbl 0923.11127.
  5. ^ Sloane, N. J. A. (ed.). "Sequence A048242 (Numbers that are not the sum of two abundant numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  6. ^ Tatersall (2005) p.144
  7. ^ Laatsch, Richard (1986). "Measuring the abundancy of integers". Mathematics Magazine. 59 (2): 84–92. doi:10.2307/2690424. ISSN 0025-570X. JSTOR 2690424. MR 0835144. Zbl 0601.10003.
  8. ^ For smallest odd integer k with abundancy index exceeding n, see Sloane, N. J. A. (ed.). "Sequence A119240 (Least odd number k such that sigma(k)/k >= n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  9. ^ Friedman, Charles N. (1993). "Sums of divisors and Egyptian fractions". Journal of Number Theory. 44 (3): 328–339. doi:10.1006/jnth.1993.1057. MR 1233293. Zbl 0781.11015. Archived from the original on 2012-02-10. Retrieved 2012-09-29.