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Number of maximal subsets of {1..n} containing no sums of distinct elements.
6

%I #14 Oct 11 2020 03:10:32

%S 1,1,1,3,3,6,11,16,20,32,53,78,107,149,206,292,391,556,782,1062,1451,

%T 1929,2564,3404,4431,5853,7672,9999,12973,16922,22194,28655,36734,

%U 47036,60375,76866,97892,123627,157008,196633,248221,311442,390859,488327,610685

%N Number of maximal subsets of {1..n} containing no sums of distinct elements.

%H Andrew Howroyd, <a href="/A326498/a326498.txt">PARI Program</a>

%e The a(1) = 1 through a(7) = 16 subsets:

%e {1} {1,2} {1,2} {1,3} {1,2,4} {1,2,4} {1,2,4}

%e {1,3} {1,2,4} {1,2,5} {1,2,5} {1,2,5}

%e {2,3} {2,3,4} {1,3,5} {1,2,6} {1,2,6}

%e {2,3,4} {1,3,5} {1,2,7}

%e {2,4,5} {1,3,6} {1,3,6}

%e {3,4,5} {1,4,6} {1,4,6}

%e {2,3,4} {1,4,7}

%e {2,3,6} {2,3,4}

%e {2,4,5} {2,4,5}

%e {2,5,6} {2,4,7}

%e {3,4,5,6} {2,5,6}

%e {1,3,5,7}

%e {2,3,6,7}

%e {3,4,5,6}

%e {3,5,6,7}

%e {4,5,6,7}

%t fasmax[y_]:=Complement[y,Union@@(Most[Subsets[#]]&/@y)];

%t Table[Length[fasmax[Select[Subsets[Range[n]],Intersection[#,Plus@@@Subsets[#,{2,n}]]=={}&]]],{n,0,10}]

%o (PARI) \\ See link for program file.

%o for(n=0, 25, print1(A326498(n), ", ")) \\ _Andrew Howroyd_, Aug 29 2019

%Y Subsets without sums of distinct elements are A151897.

%Y Maximal sum-free subsets are A121269.

%Y Subsets with sums are A326083.

%Y Maximal subsets without products of distinct elements are A325710.

%Y Maximal subsets without sums or products of distinct elements are A326025.

%Y Cf. A007865, A103580, A326117, A326495, A326497.

%K nonn

%O 0,4

%A _Gus Wiseman_, Jul 09 2019

%E a(16)-a(40) from _Andrew Howroyd_, Aug 29 2019

%E a(41)-a(44) from _Jinyuan Wang_, Oct 11 2020