### Conjecture of Pruthviraj Hajari

This generalizes Hahari's second conjecture.
Let k ≥ 2 be an even integer. The iterates x, t(x), t(t(x)),... of the mapping

t(x) | = | (k + 1)x + 1 | if x is odd |

t(x) | = | ⌈x/k⌉ | if x is even, |

where ⌈ ⌉ denotes the *ceiling* function.
Equivalently

t(x) | = | (k + 1)x+ 1 | if x is odd |

t(x) | = | x/k | if x ≡ 0 (mod k) |

t(x) | = | ⌊x/k⌋ + 1 | if x ≡ 0 (mod 2) and x ≢ 0 (mod k), |

are conjectured by Pruthviraj Hajari to eventually reach 1 if x > 0. (See MathOverflow question.)
It seems certain that if x ≠ 0, then t^{n}(x) will eventually reach one of the cycles
0 → 0, or 1 → k + 2 → 2 → 1, or -(2t + 1) → -2t(k + 1) - k → -(2t + 1), where 0 ≤ t ≤ k/2 - 1.

*Last modified 12th December 2020 *

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