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


function sumDivs (n)
    if n < 2 then return 0 end
    local sum, sr = 1, math.sqrt(n)
    for d = 2, sr do
        if n % d == 0 then
            sum = sum + d
            if d ~= sr then sum = sum + n / d end
        end
    end
    return sum
end
 
local a, d, p, Pn = 0, 0, 0
for n = 1, 20000 do
    Pn = sumDivs(n)
    if Pn > n then a = a + 1 end
    if Pn < n then d = d + 1 end
    if Pn == n then p = p + 1 end
end
print("Abundant:", a)
print("Deficient:", d)
print("Perfect:", p)