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%I #10 Oct 07 2023 11:26:19
%S 1,2,2,4,4,6,7,11,11,15,18,23,28,36,40,50,59,70,83,101,118,141,166,
%T 195,227,268,306,358,414,478
%N Number of finite sets of positive integers with greatest non-subset-sum n.
%C 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}.
%e The a(1) = 1 through a(8) = 11 sets:
%e {2} {3} {4} {5} {6} {7} {8} {9}
%e {1,3} {1,4} {2,3} {2,4} {2,5} {2,6} {2,7}
%e {1,5} {1,6} {3,4} {3,5} {3,6}
%e {1,2,5} {1,2,6} {1,7} {1,8} {4,5}
%e {1,3,4} {1,3,5} {2,3,4}
%e {1,2,7} {1,2,8} {1,9}
%e {1,2,3,8} {1,3,6}
%e {1,4,5}
%e {1,2,9}
%e {1,2,3,9}
%e {1,2,4,9}
%t nmz[y_]:=Complement[Range[Total[y]], Total/@Subsets[y]];
%t Table[Length[Select[Join@@IntegerPartitions/@Range[n,2*n], UnsameQ@@#&&Max@@nmz[#]==n&]],{n,15}]
%Y For least instead of greatest: A188431, non-strict A126796 (ranks A325781).
%Y The version counting multisets instead of sets is A366127.
%Y These sets counted by sum are A365924, strict A365831.
%Y A046663 counts partitions without a submultiset summing k, strict A365663.
%Y A325799 counts non-subset-sums of prime indices.
%Y A365923 counts partitions by number of non-subset-sums, strict A365545.
%Y Cf. A006827, A276024, A284640, A304792, A365543, A365658, A365661, A365918, A365920, A365921, A365925.
%K nonn,more
%O 1,2
%A _Gus Wiseman_, Oct 07 2023
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