cp's OEIS Frontend

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

%I A326022 #12 Jun 03 2024 13:16:44
%S A326022 1,1,1,1,2,2,2,4,8,8,8,10,14,25,40,49,62
%N A326022 Number of minimal complete subsets of {1..n} with maximum n.
%C A326022 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:
%C A326022    0 = (empty sum)
%C A326022    1 = 1
%C A326022    2 = 2
%C A326022    3 = 3
%C A326022    4 = 1 + 3
%C A326022    5 = 2 + 3
%C A326022    6 = 6
%C A326022    7 = 6 + 1
%C A326022    8 = 6 + 2
%C A326022    9 = 6 + 3
%C A326022   10 = 1 + 3 + 6
%C A326022   11 = 2 + 3 + 6
%C A326022   12 = 1 + 2 + 3 + 6
%C A326022 and the remaining numbers 13-25 are obtained by adding 13 to each of these.
%H A326022 Andrzej Kukla and Piotr Miska, <a href="https://arxiv.org/abs/2405.18225">On practical sets and A-practical numbers</a>, arXiv:2405.18225 [math.NT], 2024.
%e A326022 The a(3) = 1 through a(9) = 8 subsets:
%e A326022   {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}
%e A326022                     {1,2,4,5}  {1,2,4,6}  {1,2,4,7}  {1,2,3,5,8}  {1,2,3,5,9}
%e A326022                                                      {1,2,3,6,8}  {1,2,3,6,9}
%e A326022                                                      {1,2,3,7,8}  {1,2,3,7,9}
%e A326022                                                                   {1,2,4,5,9}
%e A326022                                                                   {1,2,4,6,9}
%e A326022                                                                   {1,2,4,7,9}
%e A326022                                                                   {1,2,4,8,9}
%t A326022 fasmin[y_]:=Complement[y,Union@@Table[Union[s,#]&/@Rest[Subsets[Complement[Union@@y,s]]],{s,y}]];
%t A326022 Table[Length[fasmin[Select[Subsets[Range[n]],Max@@#==n&&Union[Plus@@@Subsets[#]]==Range[0,Total[#]]&]]],{n,10}]
%Y A326022 Cf. A002033, A103295, A108917, A126796, A188431, A276024.
%Y A326022 Cf. A325684, A325781, A325790, A325791, A325986, A325988, A326016, A326020, A326021, A326036.
%K A326022 nonn,more
%O A326022 1,5
%A A326022 _Gus Wiseman_, Jun 04 2019