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 A326016 Number of knapsack partitions of n such that no addition of one part up to the maximum is knapsack. 10
 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 1, 1, 0, 3, 0, 0, 0, 1, 0, 8, 0, 8, 4, 3, 0, 11, 5, 3, 2, 5, 0, 29, 2, 9, 8, 20, 2 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,21 COMMENTS An integer partition is knapsack if every distinct submultiset has a different sum. The Heinz numbers of these partitions are given by A326018. LINKS EXAMPLE The initial terms count the following partitions:   15: (5,4,3,3)   21: (7,6,5,3)   21: (7,5,3,3,3)   24: (8,7,6,3)   25: (7,5,5,4,4)   27: (9,8,7,3)   27: (9,7,6,5)   27: (8,7,3,3,3,3)   31: (10,8,6,6,1)   33: (11,9,7,3,3)   33: (11,8,5,5,4)   33: (11,7,6,6,3)   33: (11,7,3,3,3,3,3)   33: (11,5,5,4,4,4)   33: (10,9,8,3,3)   33: (10,8,6,6,3)   33: (10,8,3,3,3,3,3) MATHEMATICA sums[ptn_]:=sums[ptn]=If[Length[ptn]==1, ptn, Union@@(Join[sums[#], sums[#]+Total[ptn]-Total[#]]&/@Union[Table[Delete[ptn, i], {i, Length[ptn]}]])]; ksQ[y_]:=Length[sums[Sort[y]]]==Times@@(Length/@Split[Sort[y]]+1)-1; maxks[n_]:=Select[IntegerPartitions[n], ksQ[#]&&Select[Table[Sort[Append[#, i]], {i, Range[Max@@#]}], ksQ]=={}&]; Table[Length[maxks[n]], {n, 30}] CROSSREFS Cf. A002033, A108917, A275972, A276024. Cf. A325863, A325864, A325877, A325878, A325880, A326015, A326017, A326018. Sequence in context: A147696 A001842 A216654 * A326033 A029429 A064559 Adjacent sequences:  A326013 A326014 A326015 * A326017 A326018 A326019 KEYWORD nonn,more AUTHOR Gus Wiseman, Jun 03 2019 STATUS approved

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Last modified May 10 20:33 EDT 2021. Contains 343780 sequences. (Running on oeis4.)