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# 7x±1 problem

From OeisWiki

The

function:

always converges to the

is also known as the

function starting with the first few odd numbers.

Trajectory until 1
Steps to 1

(A317753)
(1, ...)
0
(3, 20, 10, 5, 36, 18, 9, 64, 32, 16, 8, 4, 2, 1, ...)
13
(5, 36, 18, 9, 64, 32, 16, 8, 4, 2, 1, ...)
10
(7, 48, 24, 12, 6, 3, 20, 10, 5, 36, 18, 9, 64, 32, 16, 8, 4, 2, 1, ...)
18
(9, 64, 32, 16, 8, 4, 2, 1, ...)
7
(11, 76, 38, 19, 132, 66, 33, 232, 116, 58, 29, 204, 102, 51, 356, 178, 89, 624, 312, 156, 78, 39, 272, 136, 68, 34, 17, 120, 60, 30, 15, 104, 52, 26, 13, 92, 46, 23, 160, 80, 40, 20, 10, 5, 36, 18, 9, 64, 32, 16, 8, 4, 2, 1, ...)
53
(13, 92, 46, 23, 160, 80, 40, 20, 10, 5, 36, 18, 9, 64, 32, 16, 8, 4, 2, 1, ...)
19
(15, 104, 52, 26, 13, 92, 46, 23, 160, 80, 40, 20, 10, 5, 36, 18, 9, 64, 32, 16, 8, 4, 2, 1, ...)
23
(17, 120, 60, 30, 15, 104, 52, 26, 13, 92, 46, 23, 160, 80, 40, 20, 10, 5, 36, 18, 9, 64, 32, 16, 8, 4, 2, 1, ...)
27
(19, 132, 66, 33, 232, 116, 58, 29, 204, 102, 51, 356, 178, 89, 624, 312, 156, 78, 39, 272, 136, 68, 34, 17, 120, 60, 30, 15, 104, 52, 26, 13, 92, 46, 23, 160, 80, 40, 20, 10, 5, 36, 18, 9, 64, 32, 16, 8, 4, 2, 1, ...)
50
(21, 148, 74, 37, 260, 130, 65, 456, 228, 114, 57, 400, 200, 100, 50, 25, 176, 88, 44, 22, 11, 76, 38, 19, 132, 66, 33, 232, 116, 58, 29, 204, 102, 51, 356, 178, 89, 624, 312, 156, 78, 39, 272, 136, 68, 34, 17, 120, 60, 30, 15, 104, 52, 26, 13, 92, 46, 23, 160, 80, 40, 20, 10, 5, 36, 18, 9, 64, 32, 16, 8, 4, 2, 1, ...)
73
(23, 160, 80, 40, 20, 10, 5, 36, 18, 9, 64, 32, 16, 8, 4, 2, 1, ...)
16
(25, 176, 88, 44, 22, 11, 76, 38, 19, 132, 66, 33, 232, 116, 58, 29, 204, 102, 51, 356, 178, 89, 624, 312, 156, 78, 39, 272, 136, 68, 34, 17, 120, 60, 30, 15, 104, 52, 26, 13, 92, 46, 23, 160, 80, 40, 20, 10, 5, 36, 18, 9, 64, 32, 16, 8, 4, 2, 1, ...)
58
(27, 188, 94, 47, 328, 164, 82, 41, 288, 144, 72, 36, 18, 9, 64, 32, 16, 8, 4, 2, 1, ...)
20
(29, 204, 102, 51, 356, 178, 89, 624, 312, 156, 78, 39, 272, 136, 68, 34, 17, 120, 60, 30, 15, 104, 52, 26, 13, 92, 46, 23, 160, 80, 40, 20, 10, 5, 36, 18, 9, 64, 32, 16, 8, 4, 2, 1, ...)
43

A317753 Number of steps for

to reach 1 in

problem, or − 1 if 1 is never reached. (Also called the

in

problem, or 0 if never reached. (Also called the

**concerns the iterated function**

problem7 x ± 1 |

7 x ± 1 |

a (n) = 7 n + 1 if n ≡ +1 (mod 4), a (n) = 7 n − 1 if n ≡ − 1 (mod 4), otherwise a (n) = n / 2. |

- {0, 8, 1, 20, 2, 36, 3, 48, 4, 64, 5, 76, 6, 92, 7, 104, 8, 120, 9, 132, 10, 148, 11, 160, 12, 176, 13, 188, 14, 204, 15, 216, 16, 232, 17, 244, 18, 260, 19, 272, 20, 288, 21, 300, 22, 316, 23, 328, 24, 344, 25, 356, 26, 372, 27, 384, 28, 400, 29, 412, 30, ...}

n |

*trivial cycle*{1, 8, 4, 2, 1, ...} passing through 1. The*[total] number of iterations*(or*[total] number of steps*) to reach for the first time the number 1 fromn |

*total stopping time*.*Orbits*are also known as*trajectories*. The problem is similar to the 3*x*+ 1 problem.Every trajectory on negative numbers corresponds to negated trajectory on positive numbers, and vice versa. Therefore, for negative integers, the conjecture is that the sequence will always converge to the cycle passing through − 1. Obviously 0 is stuck on 0.

The following table gives the sequences resulting from iterating the7 x ± 1 |

7 x ± 1 |

n |

(A317753)

**1****3****5****7****9****11****13****15****17****19****21****23****25****27****29**n |

7 x ± 1 |

*total stopping time*.)- {0, 1, 13, 2, 10, 14, 18, 3, 7, 11, 53, 15, 19, 19, 23, 4, 27, 8, 50, 12, 73, 54, 16, 16, 58, 20, 20, 20, 43, 24, 24, 5, 47, 28, 325, 9, 70, 51, 32, 13, 13, 74, 272, 55, 55, 17, 17, 17, 276, 59, 40, 21, 40, 21, 21, 21, 63, 44, 63, ...}

n |

7 x ± 1 |

*dropping time*,*glide*, or*stopping time*.)- {0, 1, 12, 1, 8, 1, 4, 1, 4, 1, 42, 1, 8, 1, 4, 1, 4, 1, 23, 1, 20, 1, 4, 1, 4, 1, 12, 1, 16, 1, 4, 1, 4, 1, 282, 1, 12, 1, 4, 1, 4, 1, 229, 1, 50, 1, 4, 1, 4, 1, 8, 1, 35, 1, 4, 1, 4, 1, 8, 1, 50, 1, 4, 1, 4, 1, 46, 1, 8, 1, 4, 1, 4, 1, 225, 1, 8, 1, 4, 1, 4, 1, 35, 1, 16, 1, 4, 1, 4, 1, 46, 1, 27, 1, 4, 1, 4, 1, 16, ...}

## See also

## References

- David Barina, “7
*x*± 1: Close Relative of Collatz Problem,” 2018; arXiv:1807.00908. - Keith Matthews’ pages on the 7
*x*± 1 problem.