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A366129
Number of finite sets of positive integers with greatest non-subset-sum n.
1
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}.
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.
Sequence in context: A035940 A067772 A078374 * A341697 A242984 A027590
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Oct 07 2023
STATUS
approved