cp's OEIS Frontend

A384883 Number of maximal sparse subsets of the binary indices of n, where a set is sparse iff 1 is not a first difference.

%I A384883 #8 Jul 03 2025 09:29:30
%S A384883 1,1,1,2,1,1,2,2,1,1,1,2,2,2,2,3,1,1,1,2,1,1,2,2,2,2,2,4,2,2,3,4,1,1,
%T A384883 1,2,1,1,2,2,1,1,1,2,2,2,2,3,2,2,2,4,2,2,4,4,2,2,2,4,3,3,4,5,1,1,1,2,
%U A384883 1,1,2,2,1,1,1,2,2,2,2,3,1,1,1,2,1,1,2
%N A384883 Number of maximal sparse subsets of the binary indices of n, where a set is sparse iff 1 is not a first difference.
%C A384883 A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.
%e A384883 The binary indices of 27 are {1,2,4,5}, with maximal sparse subsets {{1,4},{1,5},{2,4},{2,5}}, so a(27) = 4.
%t A384883 spars[S_]:=Select[Subsets[S],FreeQ[Differences[#],1]&];
%t A384883 bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t A384883 maximize[sys_]:=Complement@@Prepend[Most[Subsets[#]]&/@sys,sys];
%t A384883 Table[Length[maximize[spars[bpe[n]]]],{n,0,100}]
%Y A384883 For subsets of {1..n} we get A000931 (shifted), maximal case of A000045 (shifted).
%Y A384883 This is the maximal case of A245564.
%Y A384883 The greatest number whose binary indices are one of these subsets is A374356.
%Y A384883 For prime instead of binary indices we have A385215, maximal case of A166469.
%Y A384883 A034839 counts subsets by number of maximal runs, for strict partitions A116674.
%Y A384883 A202064 counts subsets containing n with k maximal runs.
%Y A384883 A384877 gives lengths of maximal anti-runs in binary indices, firsts A384878.
%Y A384883 A384893 counts subsets by number of maximal anti-runs, for partitions A268193, A384905.
%Y A384883 Cf. A000071, A001629, A010049, A044813, A053538, A119900, A202023, A208342, A384177, A384890.
%K A384883 nonn
%O A384883 0,4
%A A384883 _Gus Wiseman_, Jul 02 2025