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A220182 Number of changes of parity in the Collatz trajectory of n.

%I A220182 #143 Apr 02 2025 23:15:17
%S A220182 0,1,4,1,2,5,10,1,12,3,8,5,4,11,10,1,6,13,12,3,2,9,8,5,14,5,82,11,10,
%T A220182 11,78,1,16,7,6,13,12,13,22,3,80,3,18,9,8,9,76,5,14,15,14,5,4,83,82,
%U A220182 11,20,11,20,11,10,79,78,1,16,17,16,7,6,7,74,13,84,13
%N A220182 Number of changes of parity in the Collatz trajectory of n.
%C A220182 For n < 10^10, if n <> 27, f(n) is finite, f(n) < 3n + 1. If n = 27 = 3^3, f(n) = 82 = 81 + 1 = 3^4 + 1 = 3n + 1. I conjecture that for any n <> 27, f(n) is finite, f(n) < 3n + 1. - _Sergey Pavlov_, Jun 02 2019. Note that this conjecture is stronger than the Collatz conjecture. - _Andrey Zabolotskiy_,  Jun 13 2019
%D A220182 R. K. Guy, Unsolved Problems in Number Theory, E16
%H A220182 T. D. Noe, <a href="/A220182/b220182.txt">Table of n, a(n) for n = 1..10000</a>
%H A220182 <a href="/index/3#3x1">Index entries for sequences related to 3x+1 (or Collatz) problem</a>
%F A220182 a(n) = a(A139391(n)) + (n mod 2) + 1 for n >= 2. - _Alan Michael Gómez Calderón_, Apr 01 2025
%e A220182 For n=5, Collatz trajectory for 5 is: 5,16,8,4,2,1; hence the number of transitions between odd and even parity is a(5)=2.
%e A220182 Similarly for n=11, Collatz trajectory gives 11,34,17,52,26,13,40,20,10,5,16,8,4,2,1; implies that a(11)=8.
%t A220182 Collatz[n_] := NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, n, # > 1 &]; parity[n_] := If[OddQ[n], 1, 0]; Table[p = parity /@ Collatz[n]; If[OddQ[n], 2*Total[p] - 2, 2*Total[p] - 1], {n, 100}] (* _T. D. Noe_, Feb 24 2013 *)
%o A220182 (PARI) next_iter(n) = if(n%2==0, return(n/2), return(3*n+1))
%o A220182 parity(n) = n%2
%o A220182 a(n) = my(x=n, par=parity(x), i=0); while(x > 1, x=next_iter(x); if(parity(x)!=par, i++; par=parity(x))); i \\ _Felix Fröhlich_, Jun 02 2019
%Y A220182 Cf. A006577, A006667, A139391.
%K A220182 nonn,look
%O A220182 1,3
%A A220182 _Jayanta Basu_, Feb 20 2013