These define three classifications of positive integers based on their   proper divisors.

Let   P(n)   be the sum of the proper divisors of   n   where the proper divisors are all positive divisors of   n   other than   n   itself.

   if    P(n) <  n    then  n  is classed as  deficient  (OEIS A005100).
   if    P(n) == n    then  n  is classed as  perfect    (OEIS A000396).
   if    P(n) >  n    then  n  is classed as  abundant   (OEIS A005101).



Example

6   has proper divisors of   1,   2,   and   3.

1 + 2 + 3 = 6,   so   6   is classed as a perfect number.



Task

Calculate how many of the integers   1   to   20,000   (inclusive) are in each of the three classes.

Show the results here.



Related tasks

   Aliquot sequence classifications.   (The whole series from which this task is a subset.)
   Proper divisors
   Amicable pairs


#include <iostream>
#include <algorithm>
#include <vector>
 
std::vector<int> findProperDivisors ( int n ) {
   std::vector<int> divisors ;
   for ( int i = 1 ; i < n / 2 + 1 ; i++ ) {
      if ( n % i == 0 ) 
	 divisors.push_back( i ) ;
   }
   return divisors  ;
}
 
int main( ) {
   std::vector<int> deficients , perfects , abundants , divisors ;
   for ( int n = 1 ; n < 20001 ; n++ ) {
      divisors = findProperDivisors( n ) ;
      int sum = std::accumulate( divisors.begin( ) , divisors.end( ) , 0 ) ;
      if ( sum < n ) {
	 deficients.push_back( n ) ;
      }
      if ( sum == n ) {
	 perfects.push_back( n ) ;
      }
      if ( sum > n ) {
	 abundants.push_back( n ) ;
      }
   }
   std::cout << "Deficient : " << deficients.size( ) << std::endl ;
   std::cout << "Perfect   : " << perfects.size( ) << std::endl ;
   std::cout << "Abundant  : " << abundants.size( ) << std::endl ;
   return 0 ;
}