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

1,3

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

Formula

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

a(n) = 2 * A006667(n) + 1 - (n mod 2). - V. Barbera, May 27 2026

Example

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.

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 *)

Prog

(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

Crossrefs

Cf. A006577, A006667, A139391.

Sequence in context: A087225 A366619 A343404 * A388339 A076064 A293570

Adjacent sequences: A220179 A220180 A220181 * A220183 A220184 A220185

Keyword

nonn,look

Author

Jayanta Basu, Feb 20 2013

Status

approved