Re^2: practical numbers

[Richard Mathar]

> From seqfan-owner at ext.jussieu.fr  Mon Nov 27 18:44:49 2006
> From: "David Wilson" <davidwwilson at comcast.net>
> To: "Sequence Fans" <seqfan at ext.jussieu.fr>
> Subject: Re: practical numbers
> ...
> Anyway, another question: What is the smallest number a(n) that is a sum of 
> its distinct divisors in n ways? 

For each n we get the "trivial" solution where the sum consists of n itself
as the only term. Further solutions, where the sum contains more than one term
(all of which are distinct divisors), are listed below in the format
n [list of terms that sum to n]:

6 [1, 2, 3]
12 [1, 2, 3, 6]
12 [2, 4, 6]
18 [1, 2, 6, 9]
18 [3, 6, 9]
20 [1, 4, 5, 10]
24 [1, 2, 3, 4, 6, 8]
24 [1, 2, 3, 6, 12]
24 [2, 4, 6, 12]
24 [1, 3, 8, 12]
24 [4, 8, 12]
28 [1, 2, 4, 7, 14]
30 [1, 3, 5, 6, 15]
30 [2, 3, 10, 15]
30 [5, 10, 15]
36 [2, 3, 4, 6, 9, 12]
36 [2, 3, 4, 9, 18]
36 [1, 2, 6, 9, 18]
36 [3, 6, 9, 18]
36 [1, 2, 3, 12, 18]
36 [2, 4, 12, 18]
36 [6, 12, 18]
40 [1, 2, 4, 5, 8, 20]
40 [1, 4, 5, 10, 20]
40 [2, 8, 10, 20]
42 [1, 6, 14, 21]
42 [7, 14, 21]


n=6 has 1 extra solution, n=12 or 18 have 2 extra solutions, n=20 has one,
n=24 has five etc.

Maple program to create that one (not checked!):

a := proc(n)
	local findx,d,f,found,i,count ;
	count :=0 :
	# decompose n into all sums that are candidates for the decomposition
	d := combinat[partition](n) ;
	# take each sum indivdually
	for findx from 1 to nops(d) do
		f := op(findx,d) ;
		# start with the assumption that it works
		found := true ;
		for i from 2 to nops(f) do
			# if two terms are equal in the Maple list, 
			# we flag this case as "not admitted"
			if op(i,f) = op(i-1,f) then
				found := false ;
				break ;
			fi ;
		od ;
		for i from 1 to nops(f) do
			# if any term is not a divisor,
			# we flag this case as "not admitted"
			if n mod op(i,f) <> 0 then
				found := false ;
				break ;
			fi ;
		od ;
		# If this sum seems to work and has more than one term, we list it
		if found and nops(f) > 1 then
			printf("%a %a\n",n,f) ;
			count := count+1 ;
		fi ;
	od ;
	RETURN(count) ;
end:

# loop over the most fundamental cases..
for n from 1 to 47 do
	a(n) ;
od ;

# Richard Mathar