OFFSET
1,5
COMMENTS
A set of positive integers summing to m is complete if every nonnegative integer up to m is the sum of some subset. For example, (1,2,3,6,13) is a complete set because we have:
0 = (empty sum)
1 = 1
2 = 2
3 = 3
4 = 1 + 3
5 = 2 + 3
6 = 6
7 = 6 + 1
8 = 6 + 2
9 = 6 + 3
10 = 1 + 3 + 6
11 = 2 + 3 + 6
12 = 1 + 2 + 3 + 6
and the remaining numbers 13-25 are obtained by adding 13 to each of these.
LINKS
Andrzej Kukla and Piotr Miska, On practical sets and A-practical numbers, arXiv:2405.18225 [math.NT], 2024.
EXAMPLE
The a(3) = 1 through a(9) = 8 subsets:
{1,2,3} {1,2,4} {1,2,3,5} {1,2,3,6} {1,2,3,7} {1,2,4,8} {1,2,3,4,9}
{1,2,4,5} {1,2,4,6} {1,2,4,7} {1,2,3,5,8} {1,2,3,5,9}
{1,2,3,6,8} {1,2,3,6,9}
{1,2,3,7,8} {1,2,3,7,9}
{1,2,4,5,9}
{1,2,4,6,9}
{1,2,4,7,9}
{1,2,4,8,9}
MATHEMATICA
fasmin[y_]:=Complement[y, Union@@Table[Union[s, #]&/@Rest[Subsets[Complement[Union@@y, s]]], {s, y}]];
Table[Length[fasmin[Select[Subsets[Range[n]], Max@@#==n&&Union[Plus@@@Subsets[#]]==Range[0, Total[#]]&]]], {n, 10}]
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Jun 04 2019
STATUS
approved