A326022 Number of minimal complete subsets of {1..n} with maximum n.

1, 1, 1, 1, 2, 2, 2, 4, 8, 8, 8, 10, 14, 25, 40, 49, 62

Offset: 1

Author

Gus Wiseman, Jun 04 2019

Keywords

• nonn
• more

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.

Examples

			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}
		

Links

• Andrzej Kukla and Piotr Miska, On practical sets and A-practical numbers, arXiv:2405.18225 [math.NT], 2024.

Crossrefs

Cf. A002033, A103295, A108917, A126796, A188431, A276024.

Cf. A325684, A325781, A325790, A325791, A325986, A325988, A326016, A326020, A326021, A326036.

Programs

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}]