cp's OEIS Frontend

A088528 Let m = number of ways of partitioning n into parts using all the parts of a subset of {1, 2, ..., n-1} whose sum of all parts of a subset is less than n; a(n) gives number of different subsets of {1, 2, ..., n-1} whose m is 0.

%I A088528 #19 Sep 12 2023 20:49:10
%S A088528 0,0,1,1,3,3,6,6,10,12,17,18,26,30,40,44,58,66,84,95,120,135,166,186,
%T A088528 230,257,314,350,421,476,561,626,749,831,986,1095,1276,1424,1666,1849,
%U A088528 2138,2388,2741,3042,3522,3879,4441,4928,5617,6222,7084,7802,8852,9800
%N A088528 Let m = number of ways of partitioning n into parts using all the parts of a subset of {1, 2, ..., n-1} whose sum of all parts of a subset is less than n; a(n) gives number of different subsets of {1, 2, ..., n-1} whose m is 0.
%C A088528 Note that {2, 3} is counted for n = 6 because although 6 = 2+2+2 = 3+3, there is no partition that includes both 2 and 3. - _David Wasserman_, Aug 09 2005
%C A088528 Said differently, a(n) is the number of finite nonempty sets of positive integers with sum < n that cannot be linearly combined using all positive coefficients to obtain n. - _Gus Wiseman_, Sep 10 2023
%e A088528 a(5)=3 because there are three different subsets, {2}, {3} & {4}; a(6)=3 because there are three different subsets, {4}, {5} & {2,3}.
%e A088528 From _Gus Wiseman_, Sep 10 2023: (Start)
%e A088528 The set {3,5} is not counted under a(8) because 1*3 + 1*5 = 8, but it is counted under a(9) and a(10), and it is not counted under a(11) because 2*3 + 1*5 = 11.
%e A088528 The a(3) = 1 through a(11) = 17 subsets:
%e A088528   {2}  {3}  {2}  {4}    {2}    {3}    {2}    {3}      {2}
%e A088528             {3}  {5}    {3}    {5}    {4}    {4}      {3}
%e A088528             {4}  {2,3}  {4}    {6}    {5}    {6}      {4}
%e A088528                         {5}    {7}    {6}    {7}      {5}
%e A088528                         {6}    {2,5}  {7}    {8}      {6}
%e A088528                         {2,4}  {3,4}  {8}    {9}      {7}
%e A088528                                       {2,4}  {2,5}    {8}
%e A088528                                       {2,6}  {2,7}    {9}
%e A088528                                       {3,4}  {3,5}    {10}
%e A088528                                       {3,5}  {3,6}    {2,4}
%e A088528                                              {4,5}    {2,6}
%e A088528                                              {2,3,4}  {2,8}
%e A088528                                                       {3,6}
%e A088528                                                       {3,7}
%e A088528                                                       {4,5}
%e A088528                                                       {4,6}
%e A088528                                                       {2,3,5}
%e A088528 (End)
%t A088528 combp[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,1,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];
%t A088528 Table[Length[Select[Select[Subsets[Range[n]],0<Total[#]<n&],combp[n,#]=={}&]],{n,15}] (* _Gus Wiseman_, Sep 12 2023 *)
%Y A088528 The complement is A088571, allowing sum n A088314.
%Y A088528 For sets with max < n instead of sum < n we have A365045, nonempty A070880.
%Y A088528 For nonnegative coefficients we have A365312, complement A365311.
%Y A088528 For sets with max <= n we have A365322.
%Y A088528 For partitions we have A365323, nonnegative A365378.
%Y A088528 A116861 and A364916 count linear combinations of strict partitions.
%Y A088528 A326083 and A124506 appear to count combination-free subsets.
%Y A088528 Cf. A000009, A326080, A364350, A364534, A365043, A365321, A365380.
%K A088528 easy,nonn
%O A088528 1,5
%A A088528 _Naohiro Nomoto_, Nov 16 2003
%E A088528 More terms from _David Wasserman_, Aug 09 2005