K-sequence

[y.kohmoto]

    Hello, seqfans.
    I conjectured the following sequence is divergent.

>   x(n)=[a*x(n-1)+b]/p^r,
>   a, b are real number, [x] is integer part of x,
>   p is prime, p^r is the highest power of p dividing [a*x(n-1)+b]

    x(0), p, a, b = 107, 2, 1.6, 1.1

    Don Reble calculated x(2,000,000) = 852756...564079; it has 46892
digits.
    The conjecture seems to be correct. But I don't know how to prove it.
    On the cases {1.0<b<=1.2}, I observed the same phenomena as above.

    If anyone proved it, tell me it.

    a table :
 　                           x(0)=3, p=2, b=1.1

              a -------- 1.5      1.599     1.59999      1.6
1.60001     1.601      1.7
              period ---- 1        20         172           divergent?    87
32          1

    Yasutoshi

    http://boat.zero.ad.jp/~zbi74583/another02.htm