### Example 1 of G. Venturini

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

T(x) = | 2500x/6 + 1 | if x ≡ 0 (mod 6), |

T(x) = | (21x - 9)/6 | if x ≡ 1 (mod 6), |

T(x) = | (x + 16)/6 | if x ≡ 2 (mod 6), |

T(x) = | (21x - 51)/6 | if x ≡ 3 (mod 6), |

T(x) = | (21x - 72)/6 | if x ≡ 4 (mod 6), |

T(x) = | (x + 13)/6 | if x ≡ 5 (mod 6) |

are printed and the number of steps taken to reach one of the integers
2 or 6 is recorded.

G. Venturini conjectured that every trajectory will end in one of the two cycles:

cycle 1: 2,3

cycle 2: 6, 2501, 419, 72, 30001, 105002, 17503, 61259, 10212, 4255001,
709169, 118197, 413681, 68949, 241313, 40221, 140765, 23463, 82112,
13688, 2284, 7982, 1333, 4664, 780, 325001, 54169, 189590, 31601,
5269, 18440, 3076, 10754, 1795, 6281, 1049, 177, 611, 104, 20.

See G. Venturini, *Iterates of Number Theoretic Functions with Periodic Rational Coefficients (Generalization of the 3x+1 Problem)*, Studies in Applied Mathematics, 86 (1992)185-218.

*Last modified 18th May 2006*

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