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Run Example
(%i1)f(x):= %i *(x + b + c/x);
                                                c
(%o1)                       f(x) := %i (x + b + -)
                                                x
(%i2) g(x):= f(f(x));
(%o2)                           g(x) := f(f(x))
(%i3) h(x):= g(g(x));
(%o3)                           h(x) := g(g(x))
(%i4) factor( rat( h(x)-x ));
               12         2  12             11          11         3  11
(%o4) (c (2 c x   - 2 %i b  x   - 3 %i b c x   + 9 b c x   - 9 %i b  x
      3  11      2  10          2    10       2    10          4  10
 - 3 b  x   + 8 c  x   - 19 %i b  c x   + 15 b  c x   - 15 %i b  x
       4  10            2  9         2  9          3    9    3    9
 - 11 b  x   - 10 %i b c  x  + 33 b c  x  - 48 %i b  c x  + b  c x
          5  9       5  9       3  8          2  2  8       2  2  8
 - 11 %i b  x  - 15 b  x  + 15 c  x  - 51 %i b  c  x  + 48 b  c  x
          4    8       4    8         6  8      6  8            3  7
 - 55 %i b  c x  - 27 b  c x  - 3 %i b  x  - 9 b  x  - 18 %i b c  x
         3  7          3  2  7       3  2  7          5    7       5    7
 + 51 b c  x  - 95 %i b  c  x  + 15 b  c  x  - 27 %i b  c x  - 30 b  c x
      7  7       4  6          2  3  6       2  3  6          4  2  6
 - 2 b  x  + 15 c  x  - 69 %i b  c  x  + 61 b  c  x  - 73 %i b  c  x
       4  2  6         6    6       6    6            4  5         4  5
 - 26 b  c  x  - 4 %i b  c x  - 10 b  c x  - 18 %i b c  x  + 44 b c  x
          3  3  5       3  3  5          5  2  5       5  2  5       5  4
 - 87 %i b  c  x  + 16 b  c  x  - 19 %i b  c  x  - 18 b  c  x  + 10 c  x
          2  4  4       2  4  4          4  3  4       4  3  4            5  3
 - 48 %i b  c  x  + 39 b  c  x  - 36 %i b  c  x  - 11 b  c  x  - 10 %i b c  x
         5  3          3  4  3      3  4  3      6  2          2  5  2
 + 22 b c  x  - 34 %i b  c  x  + 6 b  c  x  + 4 c  x  - 16 %i b  c  x
       2  5  2           6          6      7
 + 12 b  c  x  - 3 %i b c  x + 6 b c  x + c ))
      2              12           11        11        10          2  10      2  10              9           9          3  9    3  9      2  8          2    8       2    8          4  8       4  8            2  7         2  7          3    7    3    7         5  7      5  7       3  6          2  2  6       2  2  6          4    6       4    6      6  6            3  5         3  5          3  2  5    3  2  5         5    5      5    5      4  4          2  3  4       2  3  4          4  2  4       4  2  4            4  3         4  3          3  3  3    3  3  3      5  2          2  4  2      2  4  2           5          5      6
/(x (x  + b x + c) (x   - 3 %i b x   + 5 b x   + 4 c x   - 13 %i b  x   + 7 b  x   - 10 %i b c x  + 19 b c x  - 21 %i b  x  - b  x  + 9 c  x  - 39 %i b  c x  + 25 b  c x  - 15 %i b  x  - 10 b  x  - 18 %i b c  x  + 35 b c  x  - 52 %i b  c x  + b  c x  - 4 %i b  x  - 8 b  x  + 11 c  x  - 55 %i b  c  x  + 37 b  c  x  - 27 %i b  c x  - 17 b  c x  - 2 b  x  - 18 %i b c  x  + 35 b c  x  - 52 %i b  c  x  + b  c  x  - 4 %i b  c x  - 8 b  c x  + 9 c  x  - 39 %i b  c  x  + 25 b  c  x  - 15 %i b  c  x  - 10 b  c  x  - 10 %i b c  x  + 19 b c  x  - 21 %i b  c  x  - b  c  x  + 4 c  x  - 13 %i b  c  x  + 7 b  c  x  - 3 %i b c  x + 5 b c  x + c ))
(%i5) factor( rat( h(x)-x )), c:%i*b^2;
          8        8      8           7         7          2  6      2  6
(%o5) - (b  (2 %i x  + 6 x  + 8 %i b x  + 14 b x  + 24 %i b  x  + 2 b  x
          3  5       3  5         4  4       4  4          5  3       5  3
 + 25 %i b  x  - 15 b  x  - 3 %i b  x  - 27 b  x  - 16 %i b  x  - 12 b  x
          6  2       6  2         7        7      8
 - 12 %i b  x  + 12 b  x  + 6 %i b  x + 3 b  x - b ))
      14           13        13          2  12       2  12         3  11       3  11          4  10       4  10          5  9       5  9          6  8       6  8          7  7       7  7          8  6       8  6          9  5       9  5          10  4       10  4
/(x (x   - 3 %i b x   + 6 b x   - 11 %i b  x   + 12 b  x   - 6 %i b  x   + 19 b  x   + 15 %i b  x   + 38 b  x   + 34 %i b  x  + 31 b  x  + 62 %i b  x  - 13 b  x  + 45 %i b  x  - 42 b  x  - 13 %i b  x  - 62 b  x  - 34 %i b  x  - 31 b  x  - 38 %i b   x  + 15 b   x
         11  3       11  3          12  2       12  2         13        13
 - 6 %i b   x  + 19 b   x  + 12 %i b   x  + 11 b   x  + 3 %i b   x - 6 b   x
 - %i
  14
 b  ))
(%i6) f(x) := a*x^2 + b;
                                          2
(%o6)                          f(x) := a x  + b
(%i7) g(x) := c*x^2 + d;
                                          2
(%o7)                          g(x) := c x  + d
(%i8) f[0] : rat(x);
(%o8)/R/                               x
(%i9) g[0] : rat(1);
(%o9)/R/                               1
(%i10) f[n] := f(f[n-1]/g[n-1]) * g[n-1]^2;
                                    f
                                     n - 1   2
(%o10)                      f  := f(------) g
                             n      g        n - 1
                                     n - 1
(%i11) g[n] := g(f[n-1]/g[n-1]) * g[n-1]^2;
                                    f
                                     n - 1   2
(%o11)                      g  := g(------) g
                             n      g        n - 1
                                     n - 1
(%i12) precycle(m, n) := f[m]*g[n]-g[m]*f[n];
(%o12)                  precycle(m, n) := f  g  - g  f
                                           m  n    m  n
(%i13) cycle(n):= lreduce("/", rest( divisors(n), -1), precycle(0, n));
(%o13) cycle(n) := lreduce("/", rest(divisors(n), - 1), precycle(0, n))
(%i14) cycle[2];
(%o14)                              cycle
                                         2
(%i15) 
Run Example
divisors(48*5^n);
                                             n
(%o1)                           divisors(48 5 )
(%i2) 
Run Example
s: divisors(313);
(%o1)                              {1, 313}
(%i2)  lreduce ("+", args(s)) - 28;
(%o2)                                 286
(%i3)  divisors (a);
(%o3)                             divisors(a)
(%i4)  subst (8, a, %);
(%o4)             (%i5)  divisors (a = b);
(%o5)                      divisors(a) = divisors(b)
(%i6)  divisors ([a, b, c]);
(%o6)               [divisors(a), divisors(b), divisors(c)]
(%i7)  divisors (matrix ([a, b], [c, d]));
                         [ divisors(a)  divisors(b) ]
(%o7)                    [                          ]
                         [ divisors(c)  divisors(d) ]
(%i8)  divisors ({a, b, c});
(%o8)               {divisors(a), divisors(b), divisors(c)}
(%i9) 

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