This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A372119 #6 Apr 20 2024 10:51:33 %S A372119 1,2,4,5,6,7,8,9,12,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31, %T A372119 32,33,34,35,40,42,48,49,56,64,65,66,67,68,69,70,71,72,73,74,75,76,77, %U A372119 78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96 %N A372119 Numbers k such that the k-th composition in standard order is not biquanimous. %C A372119 The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions. %C A372119 A finite multiset of numbers is defined to be biquanimous iff it can be partitioned into two multisets with equal sums. Biquanimous partitions are counted by A002219 and ranked by A357976. %e A372119 The terms and corresponding compositions begin: %e A372119 1: (1) %e A372119 2: (2) %e A372119 4: (3) %e A372119 5: (2,1) %e A372119 6: (1,2) %e A372119 7: (1,1,1) %e A372119 8: (4) %e A372119 9: (3,1) %e A372119 12: (1,3) %e A372119 16: (5) %e A372119 17: (4,1) %e A372119 18: (3,2) %e A372119 19: (3,1,1) %e A372119 20: (2,3) %e A372119 21: (2,2,1) %e A372119 22: (2,1,2) %e A372119 23: (2,1,1,1) %t A372119 stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse; %t A372119 Select[Range[0,100],!MemberQ[Total/@Subsets[stc[#]], Total[stc[#]]/2]&] %Y A372119 The unordered complement is A357976, counted by A002219. %Y A372119 The unordered version is A371731, counted by A371795, even case A006827. %Y A372119 These compositions are counted by A371956. %Y A372119 The complement is A372120, counted by A064914. %Y A372119 A237258 (aerated) counts biquanimous strict partitions, ranks A357854. %Y A372119 A321142 and A371794 count non-biquanimous strict partitions. %Y A372119 A371791 counts biquanimous sets, differences A232466. %Y A372119 A371792 counts non-biquanimous sets, differences A371793. %Y A372119 Cf. A027187, A035470, A357879, A367094, A371781, A371782, A371783. %K A372119 nonn %O A372119 1,2 %A A372119 _Gus Wiseman_, Apr 20 2024