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


 
dim _proper_divisors(100)
 
sub proper_divisors(n)
	dim i
	dim _proper_divisors_count = 0
	if n <> 1 then
		for i = 1 to (n \ 2)
			if n %% i = 0 then
				_proper_divisors_count = _proper_divisors_count + 1
				_proper_divisors(_proper_divisors_count) = i 
			end if
		next
	end if
	return _proper_divisors_count
end sub
 
sub show_proper_divisors(n, tabbed)
	dim cnt = proper_divisors(n)
	print str$(n) + ":"; tab(4);"(" + str$(cnt) + " items) ";
	dim j 
	for j = 1 to cnt
		if tabbed then
			print str$(_proper_divisors(j)),
		else 
			print str$(_proper_divisors(j));
		end if
		if (j < cnt) then print ",";
	next
	print
end sub
 
dim i
for i = 1 to 10
    show_proper_divisors(i, false)
next
 
dim c 
dim maxindex = 0
dim maxlength = 0
for t = 1 to 20000
	c = proper_divisors(t)
	if c > maxlength then
		maxindex = t
		maxlength = c
	end if
next
 
print "A maximum at ";
show_proper_divisors(maxindex, false)
 