Implementing Goodstein Sequences in J

I recently read about Goodstein sequences for the first time, and was interested in writing a J verb that can generate a sequence. Goodstein sequences are defined here: definition A Goodstein sequence is defined in terms of hereditary representations of numbers. An example of hereditary representation of 100 using base 2. \[ 100 = 2^{6}+2^{5}+2^{2} = 2^{2^{2}+2}+2^{2^{2}+1}+2^{2}.\] Let the hereditary representation of \(n\) using base \(2\) be \(G_{0}(n)\), i.e. the start of the sequence. Then \(G_{1}(n)\) uses the hereditary representation of \(n\) base 2, but exchanges 3 for 2 in all places in this representation. Finally, we minus 1. We can continue, so in general, \[G_{m+1}(n) = H_{m+1}(G_{m}) - 1\] where \(H_{a}(x)\) is the base \(a\) hereditary representation of \(x\). For example: \[G_{0}(100) = 100 = 2^{6}+2^{5}+2^{2} = 2^{2^{2}+2}+2^{2^{2}+1}+2^{2}.\] \[G_{1}(100) = 3^{3^{3}+3}+3^{3^{3}+1}+3^{3} - 1 = 228767924549636.\] \[G_{2}(100) = 34860300617850752458892464995335199931446351133540222782081259753676586478191222...\] \[G_{3}(100) = 59814694315693446387155462718734091954899936833389952752466750303416092596593874...\] This sequence grows explosively fast for most values of \(n\). Interestingly, Goodstein's Theorem states that it will always terminate at 0 eventually. For all positive values of \(n\). Even though it is futile to attempt computing a Goodstein sequence for most values of \(n\), as the numbers involved become unmanageable very quickly, it is still interesting and worthwhile writing a verb to do such a thing. Here is one implementation in J:

goodstein=: 4 : 0"0 0

if. y <: 0 do. 0 return.
elseif. y = x do. x+1 return.
end.  
s=. I. (x:x) (|.@:((>:@:>.@:^. # [) #: ])) x:y
+/ (x+1) ^ (x: x) goodstein s
)

G=: <:@:goodstein`]@.(0&=@:])

NB. generates sequence
genSeq=: 3 : 0"1
'base val its'=. y
c=. 0
vals=. val
whilst. its > c=. c+1 do.
  val=. base G val
  vals=. vals,val
  base=. base+1
  if. val = 0 do. break.end.
end.
vals
)

There are three verbs here, goodstein, G, genSeq. goodstein is a recursive dyadic verb which calculates the hereditary representation of the right argument, with base the left argument, and then exchanges occurences of the base by \(base+1\). The reason for the recursion is that for each exponent of the base in the hereditary representation sum, we also need to calculate a hereditary representation. G merely subtracts one from the return value of goodstein.FInally, genSeq, is used to simply generate a sequence. It takes three arguments; the start base (which should be 2); the starting value; the number of iterations to calculate. Example:

genSeq 2 100 3

will return 100 228767924549636 34860300617850752458892464995...