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Finite Sequence Generator

Published on Saturday, 14th December 2019, 04:00 pm; Solved by 212;
Difficulty rating: 40%

Problem 693

Two positive integers $x$ and $y$ ($x > y$) can generate a sequence in the following manner:

  • $a_x = y$ is the first term,
  • $a_{z+1} = a_z^2 \bmod z$ for $z = x, x+1,x+2,\ldots$ and
  • the generation stops when a term becomes 0 or 1.

The number of terms in this sequence is denoted $l(x,y)$.

For example, with $x = 5$ and $y = 3$, we get $a_5 = 3$, $a_6 = 3^2 \bmod 5 = 4$, $a_7 = 4^2\bmod 6 = 4$, etc. Giving the sequence of 29 terms:
$ 3,4,4,2,4,7,9,4,4,3,9,6,4,16,4,16,16,4,16,3,9,6,10,19,25,16,16,8,0 $
Hence $l(5,3) = 29$.

$g(x)$ is defined to be the maximum value of $l(x,y)$ for $y < x$. For example, $g(5) = 29$.

Further, define $f(n)$ to be the maximum value of $g(x)$ for $x \le n$. For example, $f(100) = 145$ and $f(10\,000) = 8824$.

Find $f(3\,000\,000)$.