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A333629 Least k such that the runs-resistance of the k-th composition in standard order is n.

%I A333629 #11 Aug 05 2025 03:38:56
%S A333629 1,3,5,11,27,93,859,13789,1530805,1567323995
%N A333629 Least k such that the runs-resistance of the k-th composition in standard order is n.
%C A333629 A composition of n is a finite sequence of positive integers summing to n. The k-th composition in standard order (row k of 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.
%C A333629 For the operation of taking the sequence of run-lengths of a finite sequence, runs-resistance is defined as the number of applications required to reach a singleton.
%H A333629 Claude Lenormand, <a href="/A318921/a318921.pdf">Deux transformations sur les mots</a>, Preprint, 5 pages, Nov 17 2003.
%e A333629 The sequence together with the corresponding compositions begins:
%e A333629         1: (1)
%e A333629         3: (1,1)
%e A333629         5: (2,1)
%e A333629        11: (2,1,1)
%e A333629        27: (1,2,1,1)
%e A333629        93: (2,1,1,2,1)
%e A333629       859: (1,2,2,1,2,1,1)
%e A333629     13789: (1,2,2,1,1,2,1,1,2,1)
%e A333629   1530805: (2,1,1,2,2,1,2,1,1,2,1,2,2,1)
%e A333629 For example, starting with 13789 and repeatedly applying A333627 gives: 13789 -> 859 -> 110 -> 29 -> 11 -> 6 -> 3 -> 2, corresponding to the compositions: (1,2,2,1,1,2,1,1,2,1) -> (1,2,2,1,2,1,1) -> (1,2,1,1,2) -> (1,1,2,1) -> (2,1,1) -> (1,2) -> (1,1) -> (2).
%t A333629 nn=1000;
%t A333629 stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A333629 stcrun[n_]:=Total[2^(Accumulate[Reverse[Length/@Split[stc[n]]]])]/2;
%t A333629 seq=Table[Length[NestWhileList[stcrun,n,Length[stc[#]]>1&]]-1,{n,nn}];
%t A333629 Table[Position[seq,i][[1,1]],{i,Union[seq]}]
%Y A333629 Positions of first appearances in A333628 = number of times applying A333627 to reach a power of 2, starting with n.
%Y A333629 A subsequence of A333630.
%Y A333629 All of the following pertain to compositions in standard order (A066099):
%Y A333629 - The length is A000120.
%Y A333629 - The partial sums from the right are A048793.
%Y A333629 - The sum is A070939.
%Y A333629 - Adjacent equal pairs are counted by A124762.
%Y A333629 - Equal runs are counted by A124767.
%Y A333629 - Strict compositions are ranked by A233564.
%Y A333629 - The partial sums from the left are A272020.
%Y A333629 - Constant compositions are ranked by A272919.
%Y A333629 - Normal compositions are ranked by A333217.
%Y A333629 - Heinz number is A333219.
%Y A333629 - Anti-runs are counted by A333381.
%Y A333629 - Adjacent unequal pairs are counted by A333382.
%Y A333629 Cf. A029931, A098504, A114994, A225620, A228351, A238279, A242882, A318928, A329744, A329747, A333489.
%K A333629 nonn,more
%O A333629 0,2
%A A333629 _Gus Wiseman_, Mar 31 2020
%E A333629 a(9) from _Amiram Eldar_, Aug 04 2025