%I #22 Jan 13 2022 18:44:07
%S 1,1,2,2,4,8,10,12,17,34,45,77,99,136,166,200,238,328,402,660,674,
%T 1166,1331,1966,2335,3286,3527,4762,5383,6900,7543,9087,10149,12239,
%U 13569,16452,17867,22869,23977,33881,33820,43423,48090,68683,67347,95176,97917,131666,136205
%N Number of maximal subsets of {1..n} containing n such that every subset has a different sum.
%C These are maximal strict knapsack partitions (A275972, A326015) organized by maximum rather than sum.
%H Fausto A. C. Cariboni, <a href="/A325867/b325867.txt">Table of n, a(n) for n = 1..150</a> (terms 1..121 from Bert Dobbelaere)
%e The a(1) = 1 through a(8) = 12 subsets:
%e {1} {1,2} {1,3} {1,2,4} {1,2,5} {1,2,6} {1,2,7} {1,3,8}
%e {2,3} {2,3,4} {1,3,5} {1,3,6} {1,3,7} {1,5,8}
%e {2,4,5} {1,4,6} {1,4,7} {5,7,8}
%e {3,4,5} {2,3,6} {1,5,7} {1,2,4,8}
%e {2,5,6} {2,3,7} {1,4,6,8}
%e {3,4,6} {2,4,7} {2,3,4,8}
%e {3,5,6} {2,6,7} {2,4,5,8}
%e {4,5,6} {4,5,7} {2,4,7,8}
%e {4,6,7} {3,4,6,8}
%e {3,5,6,7} {3,6,7,8}
%e {4,5,6,8}
%e {4,6,7,8}
%t fasmax[y_]:=Complement[y,Union@@(Most[Subsets[#]]&)/@y];
%t Table[Length[fasmax[Select[Subsets[Range[n]],MemberQ[#,n]&&UnsameQ@@Plus@@@Subsets[#]&]]],{n,15}]
%o (Python)
%o def f(p0, n, m, cm):
%o full, t, p = True, 0, p0
%o while p<n:
%o sm = m<<p
%o if (m & sm) == 0:
%o t += f(p+1, n, m|sm, cm|(1<<p))
%o full=False
%o p+=1
%o if full:
%o for k in range(1, p0):
%o if ((cm>>k)&1)==0 and ((m<<k)&m)==0:
%o full=False
%o break
%o return 1 if full else t
%o def a325867(n):
%o return f(1, n, (1<<n)+1, 0)
%o # _Bert Dobbelaere_, Mar 07 2021
%Y Cf. A002033, A108917, A143823, A143824, A196723, A275972.
%Y Cf. A325860, A325861, A325864, A325865, A325866, A325867, A325880.
%K nonn
%O 1,3
%A _Gus Wiseman_, Jun 01 2019
%E More terms from _Bert Dobbelaere_, Mar 07 2021