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A335834 Sort the run lengths in binary expansion of n in ascending order (with multiplicities).

%I A335834 #19 Jun 28 2020 16:20:30
%S A335834 0,1,2,3,4,5,4,7,8,11,10,11,12,11,8,15,16,23,20,19,20,21,20,23,24,19,
%T A335834 20,19,24,23,16,31,32,47,40,39,44,43,44,39,40,43,42,43,44,43,40,47,48,
%U A335834 39,44,51,44,43,44,39,56,39,40,39,48,47,32,63,64,95,80,79
%N A335834 Sort the run lengths in binary expansion of n in ascending order (with multiplicities).
%C A335834 This sequence preserves the number of runs (A005811) and the length (A070939) of the binary representation of a number.
%H A335834 Rémy Sigrist, <a href="/A335834/b335834.txt">Table of n, a(n) for n = 0..8191</a>
%H A335834 <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%F A335834 a(a(n)) = a(n).
%e A335834 For n = 72:
%e A335834 - the binary representation of 72 is "1001000",
%e A335834 - the corresponding run lengths are: 1, 2, 1, 3,
%e A335834 - in ascending order: 1, 1, 2, 3,
%e A335834 - so the binary representation of a(72) is "1011000",
%e A335834 - and a(72) = 88.
%t A335834 Array[FromDigits[Flatten@ MapIndexed[ConstantArray[Mod[First[#2], 2], #1] &, Sort[Length /@ Split[IntegerDigits[#, 2]]]], 2] &, 67] (* _Michael De Vlieger_, Jun 27 2020 *)
%o A335834 (PARI) torl(n) = { my (rr=[]); while (n, my (r=valuation(n+(n%2), 2)); rr = concat(r, rr); n\=2^r); rr }
%o A335834 fromrl(rr) = { my (v=0); for (k=1, #rr, v = (v+(k%2))*2^rr[k]-(k%2)); v }
%o A335834 a(n) = { fromrl(vecsort(torl(n))) }
%Y A335834 Cf. A005811, A037014 (fixed points), A070939, A101211, A335835.
%K A335834 nonn,base,look,easy
%O A335834 0,3
%A A335834 _Rémy Sigrist_, Jun 26 2020