The   proper divisors   of a positive integer N are those numbers, other than N itself, that divide N without remainder.

For N > 1 they will always include 1,   but for N == 1 there are no proper divisors.



Examples

The proper divisors of     6     are   1, 2, and 3.

The proper divisors of   100   are   1, 2, 4, 5, 10, 20, 25, and 50.



Task

 Create a routine to generate all the proper divisors of a number.
 use it to show the proper divisors of the numbers 1 to 10 inclusive.
 Find a number in the range 1 to 20,000 with the most proper divisors. Show the number and just the count of how many proper divisors it has.


Show all output here.



Related tasks

   Amicable pairs
   Abundant, deficient and perfect number classifications
   Aliquot sequence classifications
   Factors of an integer
   Prime decomposition


FUNCTION CountProperDivisors (number)
    IF number < 2 THEN CountProperDivisors = 0
    count = 0
    FOR i = 1 TO number \ 2
        IF number MOD i = 0 THEN count = count + 1
    NEXT i
    CountProperDivisors = count
END FUNCTION
 
SUB ListProperDivisors (limit)
    IF limit < 1 THEN EXIT SUB
    FOR i = 1 TO limit
        PRINT USING "## ->"; i;
        IF i = 1 THEN PRINT " (None)";
        FOR j = 1 TO i \ 2
            IF i MOD j = 0 THEN PRINT " "; j;
        NEXT j
        PRINT
    NEXT i
END SUB
 
most = 1
maxCount = 0
 
PRINT "The proper divisors of the following numbers are: "; CHR$(10)
ListProperDivisors (10)
 
FOR n = 2 TO 20000
'' It is extremely slow in this loop 
    count = CountProperDivisors(n)
    IF count > maxCount THEN
        maxCount = count
        most = n
    END IF
NEXT n
 
PRINT
PRINT most; " has the most proper divisors, namely "; maxCount
END