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A365045 Number of subsets of {1..n} containing n such that no element can be written as a positive linear combination of the others.

%I A365045 #17 Dec 13 2024 09:37:26
%S A365045 0,1,1,2,4,11,23,53,111,235,483,988,1998,4036,8114,16289,32645,65389,
%T A365045 130887,261923,524014,1048251,2096753,4193832,8388034,16776544,
%U A365045 33553622,67107919,134216597,268434140,536869355,1073740012,2147481511,4294964834,8589931700
%N A365045 Number of subsets of {1..n} containing n such that no element can be written as a positive linear combination of the others.
%C A365045 Also subsets of {1..n} containing n whose greatest element cannot be written as a positive linear combination of the others.
%H A365045 Steven R. Finch, <a href="/A066062/a066062.pdf">Monoids of natural numbers</a>, March 17, 2009.
%F A365045 a(n) = A070880(n) + 1 for n > 0.
%e A365045 The subset {3,4,10} has 10 = 2*3 + 1*4 so is not counted under a(10).
%e A365045 The a(0) = 0 through a(5) = 11 subsets:
%e A365045   .  {1}  {2}  {3}    {4}        {5}
%e A365045                {2,3}  {3,4}      {2,5}
%e A365045                       {2,3,4}    {3,5}
%e A365045                       {1,2,3,4}  {4,5}
%e A365045                                  {2,4,5}
%e A365045                                  {3,4,5}
%e A365045                                  {1,2,3,5}
%e A365045                                  {1,2,4,5}
%e A365045                                  {1,3,4,5}
%e A365045                                  {2,3,4,5}
%e A365045                                  {1,2,3,4,5}
%t A365045 combp[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,1,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];
%t A365045 Table[Length[Select[Subsets[Range[n]],MemberQ[#,n]&&And@@Table[combp[#[[k]],Union[Delete[#,k]]]=={},{k,Length[#]}]&]],{n,0,10}]
%Y A365045 The nonempty case is A070880.
%Y A365045 The nonnegative version is A124506, first differences of A326083.
%Y A365045 The binary version is A288728, first differences of A007865.
%Y A365045 A subclass is A341507.
%Y A365045 The complement is counted by A365042, first differences of A365043.
%Y A365045 First differences of A365044.
%Y A365045 The nonnegative complement is A365046, first differences of A364914.
%Y A365045 The binary complement is A365070, first differences of A093971.
%Y A365045 Without re-usable parts we have A365071, first differences of A151897.
%Y A365045 A085489 and A364755 count subsets w/o the sum of two distinct elements.
%Y A365045 A088809 and A364756 count subsets with the sum of two distinct elements.
%Y A365045 A364350 counts combination-free strict partitions, complement A364839.
%Y A365045 A364913 counts combination-full partitions.
%Y A365045 Cf. A237113, A237668, A308546, A324736, A326020, A326080, A364272, A364349, A364534, A365069.
%K A365045 nonn
%O A365045 0,4
%A A365045 _Gus Wiseman_, Aug 24 2023