Aliquot sum

In <a href="/facts/Number_theory/zhoa7zrc">number theory</a>, the aliquot sum s(n) of a <a href="/facts/Positive_integer/u65og8JE">positive integer</a> n is the sum of all <a href="/facts/Proper_divisors/DKNa2GvC">proper divisors</a> of n, that is, all divisors of n other than n itself.
That is,

s
        (
        n
        )
        =
        
          ∑
          
            
              
                d
                
                  |
                
                n
                ,
              
              
                d
                ≠
                n
              
            
          
        
        d
        
        .
      
    
    {\displaystyle s(n)=\sum _{{d|n,} \atop {d\neq n}}d\,.}

It can be used to characterize the <a href="/facts/Prime_number/L20FLkgn">prime numbers</a>, <a href="/facts/Perfect_number/WyqWtApV">perfect numbers</a>, <a href="/facts/Sociable_number/1typvFxR">sociable numbers</a>, <a href="/facts/Deficient_number/dreFgv8A">deficient numbers</a>, <a href="/facts/Abundant_number/K54Q9owk">abundant numbers</a>, and <a href="/facts/Untouchable_number/RWtqyP39">untouchable numbers</a>, and to define the <a href="/facts/Aliquot_sequence/XYI13Wne">aliquot sequence</a> of a number.

Aliquot sum open-in-new

Aliquot sum