### A 6 branch generalized 3x+1 conjecture

The iterates y, t(y), t(t(y)),... of the function

t(x) | = | x/6 | if x ≡ 0 (mod 6), |

t(x) | = | (7x+1)/2 | if x ≡ 1 (mod 6), |

t(x) | = | x/2 | if x ≡ 2 (mod 6), |

t(x) | = | x/3 | if x ≡ 3 (mod 6), |

t(x) | = | x/2 | if x ≡ 4 (mod 6), |

t(x) | = | (7x+1)/6 | if x ≡ 5 (mod 6), |

are printed and the number of steps taken to reach one of the integers
19, 1, 0, -1, -5, -11 is recorded.
This mapping is of type (b) in *Generalized 3x+1 mappings* with Markov matrix

1/6 | 1/6 | 1/6 | 1/6 | 1/6 | 1/6 |

0 | 1/2 | 1/2 | 0 | 0 | 0 |

0 | 0 | 0 | 1/2 | 1/2 | 0 |

0 | 1/2 | 1/2 | 0 | 0 | 0 |

1/6 | 1/6 | 1/6 | 1/6 | 1/6 | 1/6 |

0 | 1/3 | 0 | 0 | 1/3 | 1/3 |

with stationary vector (2/53, 15/53, 14/53, 9/53, 10/53, 3/53).

Then, as we have the inequality
(1/6)^{2/53}(21/6)^{15/53}(3/6)^{14/53}(2/6)^{9/53}(3/6)^{10/53}(7/6)^{3/53} < 1,

it is conjectured by Keith Matthews that every trajectory will end in one of the numbers in this list and subsequently cycle. (The cycle lengths are printed in bold type.):

- 0 (
**1**)
- 1, 4, 2 (
**3**)
- 19, 67, 235, 823, 2881, 10084, 5042, 2521, 8824, 4412, 2206, 1103, 1287, 429, 143, 167, 195, 65, 76, 38 (
**20**)
- -5, -17, -59, -206, -103, -120, -20, -10 (
**8**)
- -11, -38, -19, -22 (
**4**)

This mapping was communicated on July 16, 2017 by Kevin Lamoreau, who was inspired by Tomás Oliveira e Silva's 5x + 1 and 7x + 1 variants.

*Last modified 17th July 2017 *

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