The Goodstein Sequence
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We define the hereditary basen representation of a number as follow:
For instance, if $n = 266$, and $base = 2$, its hereditary base2 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 base2 representation of n, replace each 2s with 3s and substract 1
$G_k(n) = $ Take the hereditary basek representation of $G_{k1}(n)$, 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 $G_{50}(13)$
[My timing: < 1 sec]
For instance, if $n = 266$, and $base = 2$, its hereditary base2 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 base2 representation of n, replace each 2s with 3s and substract 1
$G_k(n) = $ Take the hereditary basek representation of $G_{k1}(n)$, 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 $G_{50}(13)$
[My timing: < 1 sec]
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