I recently stumbled across this page and was surprised as how little content it held.
As it happens, I had a general recursive function evaluator lying around, so I went and quickly wrote a program to generate all GRF’s of a given size and arity. Combining them produced this table of how many 0-arity GRF’s of a particular size evaluated to a particular value:
| size | ∞ | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|---|
| 2 | 1 | 3 | ||||
| 3 | 1 | 9 | 1 | |||
| 4 | 8 | 39 | 4 | |||
| 5 | 38 | 207 | 17 | 2 | ||
| 6 | 217 | 1217 | 95 | 9 | ||
| 7 | 1455 | 7705 | 571 | 47 | 5 | |
| 8 | 9765 | 51601 | 3623 | 308 | 24 | |
| 9 | 69659 | 362259 | 24205 | 2017 | 146 | 14 |
The “∞” indicates functions which my evaluator failed on because their μ‒recursion produced an infinite loop. I’ve shown this to be the case for size up to 7. I haven’t for 8 and 9, but I’m sure it’s also true for them because the values produced by size m GRF’s aren’t going to suddenly jump to the trillions with no functions that generate values in the [5, 10⁹] range. I think it’s far more likely there’s a bug in the generator I hacked together than that one of the size 8 or 9 results is secretly a large number, but I appreciate that people place value in explicitly showing these things, so I only claim credit up to size 7.
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