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

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, 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, 11, 20, 11, 20, 11, 10, 79, 78, 1, 16, 17, 16, 7, 6, 7, 74, 13, 84, 13

Offset: 1

Author

Jayanta Basu, Feb 20 2013

Keywords

• nonn
• look

Comments

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

Examples

			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.
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.
		

References

• R. K. Guy, Unsolved Problems in Number Theory, E16

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000
• Index entries for sequences related to 3x+1 (or Collatz) problem

Crossrefs

Cf. A006577, A006667, A139391.

Programs

Mathematica

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 *)

PARI

next_iter(n) = if(n%2==0, return(n/2), return(3*n+1))
parity(n) = n%2
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

Formula

a(n) = a(A139391(n)) + (n mod 2) + 1 for n >= 2. - Alan Michael Gómez Calderón, Apr 01 2025