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%I #137 Dec 11 2022 01:24:20
%S 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 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 11,20,11,20,11,10,79,78,1,16,17,16,7,6,7,74,13,84,13
%N Number of changes of parity in the Collatz trajectory of n.
%C 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 R. K. Guy, Unsolved Problems in Number Theory, E16
%H T. D. Noe, <a href="/A220182/b220182.txt">Table of n, a(n) for n = 1..10000</a>
%H <a href="/index/3#3x1">Index entries for sequences related to 3x+1 (or Collatz) problem</a>
%e For n=5, Collatz trajectory for 5 is: 5,16,8,4,2,1; hence no. of transitions between odd and even parity is a(5)=2; 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 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 (PARI) next_iter(n) = if(n%2==0, return(n/2), return(3*n+1))
%o parity(n) = n%2
%o 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 Cf. A006577, A006667.
%K nonn,look
%O 1,3
%A _Jayanta Basu_, Feb 20 2013
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