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A325865 Number of maximal subsets of {1..n} of which every subset has a different sum.

%I A325865 #10 Mar 23 2025 23:40:15
%S A325865 1,1,1,3,3,6,14,23,27,40,64,104,180,275,399,554,679,872,1117,1431,
%T A325865 1920,2520,3530,4751,6644,8855,12021,15461,19939,25109,31656,38750,
%U A325865 46204,55650,65942,78045,91304,106592,124761,145701,172343,201217,238739,280601,339746,400394
%N A325865 Number of maximal subsets of {1..n} of which every subset has a different sum.
%H A325865 Andrew Howroyd, <a href="/A325865/b325865.txt">Table of n, a(n) for n = 0..60</a>
%e A325865 The a(1) = 1 through a(6) = 14 subsets:
%e A325865   {1}  {1,2}  {1,2}  {1,3}    {1,2,4}  {1,2,4}
%e A325865               {1,3}  {1,2,4}  {1,2,5}  {1,2,5}
%e A325865               {2,3}  {2,3,4}  {1,3,5}  {1,2,6}
%e A325865                               {2,3,4}  {1,3,5}
%e A325865                               {2,4,5}  {1,3,6}
%e A325865                               {3,4,5}  {1,4,6}
%e A325865                                        {2,3,4}
%e A325865                                        {2,3,6}
%e A325865                                        {2,4,5}
%e A325865                                        {2,5,6}
%e A325865                                        {3,4,5}
%e A325865                                        {3,4,6}
%e A325865                                        {3,5,6}
%e A325865                                        {4,5,6}
%t A325865 fasmax[y_]:=Complement[y,Union@@(Most[Subsets[#]]&)/@y];
%t A325865 Table[Length[fasmax[Select[Subsets[Range[n]],UnsameQ@@Plus@@@Subsets[#]&]]],{n,0,10}]
%o A325865 (PARI)
%o A325865 a(n)={
%o A325865   my(ismaxl(w)=for(k=1, n, if(!bitand(w,w<<k), return(0))); 1);
%o A325865   my(recurse(k,b,w)=
%o A325865       if(k > n, ismaxl(w),
%o A325865          my(s=self()(k+1, b,w));
%o A325865          if(!bitand(w,w<<k), s+=self()(k+1, b+(1<<k), w + (w<<k)));
%o A325865          s);
%o A325865   );
%o A325865   recurse(1,0,1);
%o A325865 } \\ _Andrew Howroyd_, Mar 23 2025
%Y A325865 Cf. A002033, A108917, A143823, A196723, A275972.
%Y A325865 Cf. A325860, A325864, A325866, A325867, A325877, A325878, A325879, A325880.
%K A325865 nonn
%O A325865 0,4
%A A325865 _Gus Wiseman_, Jun 01 2019
%E A325865 a(18) onwards from _Andrew Howroyd_, Mar 23 2025