A366129 Number of finite sets of positive integers with greatest non-subset-sum n.

1, 2, 2, 4, 4, 6, 7, 11, 11, 15, 18, 23, 28, 36, 40, 50, 59, 70, 83, 101, 118, 141, 166, 195, 227, 268, 306, 358, 414, 478

Offset

1,2

Comments

A non-subset-sum of a set summing to n is a positive integer up to n that is not the sum of any subset. For example, the non-subset-sums of {1,3,4} are {2,6}.

Links

Table of n, a(n) for n=1..30.

Example

The a(1) = 1 through a(8) = 11 sets:
  {2}  {3}    {4}    {5}      {6}      {7}      {8}        {9}
       {1,3}  {1,4}  {2,3}    {2,4}    {2,5}    {2,6}      {2,7}
                     {1,5}    {1,6}    {3,4}    {3,5}      {3,6}
                     {1,2,5}  {1,2,6}  {1,7}    {1,8}      {4,5}
                                       {1,3,4}  {1,3,5}    {2,3,4}
                                       {1,2,7}  {1,2,8}    {1,9}
                                                {1,2,3,8}  {1,3,6}
                                                           {1,4,5}
                                                           {1,2,9}
                                                           {1,2,3,9}
                                                           {1,2,4,9}

Mathematica

nmz[y_]:=Complement[Range[Total[y]], Total/@Subsets[y]];
Table[Length[Select[Join@@IntegerPartitions/@Range[n, 2*n], UnsameQ@@#&&Max@@nmz[#]==n&]], {n, 15}]

Crossrefs

For least instead of greatest: A188431, non-strict A126796 (ranks A325781).

The version counting multisets instead of sets is A366127.

These sets counted by sum are A365924, strict A365831.

A046663 counts partitions without a submultiset summing k, strict A365663.

A325799 counts non-subset-sums of prime indices.

A365923 counts partitions by number of non-subset-sums, strict A365545.

Cf. A006827, A276024, A284640, A304792, A365543, A365658, A365661, A365918, A365920, A365921, A365925.

Sequence in context: A035940 A067772 A078374 * A341697 A242984 A027590

Adjacent sequences: A366126 A366127 A366128 * A366130 A366131 A366132

Keyword

nonn,more

Author

Gus Wiseman, Oct 07 2023

Status

approved