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Problem #396

The Goodstein Sequence
 Public ★(x17) 05/19/17 by Philippe_57721 7xp Programming 85.0%

We define the hereditary base-n representation of a number as follow:
For instance, if $n = 266$, and $base = 2$, its hereditary base-2 representation is:
$266 = 2^1+2^3+2^8$
$\quad = 2^1+2^{1+2}+2^{2^3}$
$\quad = 2^1+2^{1+2}+2^{2^{1+2}}$
We express n in base 2 and recursively every exponent in base 2.

Let's define the $G_k$ sequence as follow:
$G_1(n) = n$
$G_2(n) =$ Take the hereditary base-2 representation of n, replace each 2s with 3s and substract 1
$G_k(n) =$ Take the hereditary base-k representation of $G_{k-1}(n)$, replace each $k$s with $(k+1)$s and substract 1

Here the first values for $G_k(6)$
$G_1(6) = 6$
$G_2(6) = 29$
$G_3(6) = 257$
$G_4(6) = 3125$
$G_5(6) = 46655$
$G_6(6) = 98039$
$G_7(6) = 187243$
$G_8(6) = 332147$

Find $G_{50}(13)$

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