## FANDOM

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$R_{a}^{b}(n)$ : replace $a$ to $b$ in base-$a$ hereditary representation of $n$

$R_{a}^{b}(n)=R_{a}^{b}(n,0)$
$R_{a}^{b}(0,t)=0$
$R_{a}^{b}(n,t)=R_{a}^{b}(\lfloor n/a \rfloor,t+1)+b^{R_{a}^{b}(t,0)}(n-a\lfloor n/a \rfloor))$

$g_{\alpha,n}$ : Goodstein sequence starting with $n$ and bumping base $b$ to $G_{\alpha}(b)$

$g_{\alpha,n}(0)=n$
$g_{\alpha,n}(m+1)=max(R_{{G_{\alpha}}^{m}(2)}^{{G_{\alpha}}^{m+1}(2)}(g_{\alpha,n}(m)),1)-1$

$G_{0}(n)=n+1$

$G_{\alpha+1}(n)$ : the base when $g_{\alpha,n}$ terminates

$G_{\alpha+1}(n)={G_{\alpha}}^{\mu k(g_{\alpha,n}(k)=0)}(2)$

$G_{\alpha}(n)=G_{\alpha \left [ n \right ]}(n)$ if $\alpha$ is a limit ordinal

example:

$R_{2}^{5}(11)=R_{2}^{5}(2^{2+1}+2+1)=5^{5+1}+5+1=15631$
$G_{1}(3)=7$
$n$ base $g_{0,3}(n)$
$0$ $2$ $2+1$
$1$ $3$ $3$
$2$ $4$ $3$
$3$ $5$ $2$
$4$ $6$ $1$
$5$ $7$ $0$
$G_{2}(2)={G_{1}}^{5}(2)$
$n$ base $g_{1,2}(n)$
$0$ $2$ $2$
$1$ $G_{1}(2)=5$ $4$
$2$ ${G_{1}}^{2}(2)$ $3$
$3$ ${G_{1}}^{3}(2)$ $2$
$4$ ${G_{1}}^{4}(2)$ $1$
$5$ ${G_{1}}^{5}(2)$ $0$