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This is a function that I thought of while trying to think about functions,that give out an incredibly large output unexpectedly.It is very similar to Beklemishev's worms.

Here are the rules:

If there is only one entry,then...

\(a:\{b\}=a^b\) for \(a > 0,b > 0\)

If there are more than one entries,then...

\(a:\{\underbrace{0,0,\ldots,0,0}_n,c,\ldots,d\}=(a+n):\{c,\ldots,d+n\}\)

\(a:\{b,c,\ldots,n\}=(a+1):\{\underbrace{b-1,b-1,b-1,\ldots,b-1,b-1}_{a+2},c,\ldots,n+1\}\) for \(a > 1\)

Otherwise:If \(a=0\),then \(0:\{b\}=1:\{b-1,b-1\}=2:\{b-2,b-2,b-2,b\}=3:\{b-3,b-3,b-3,b-3,b-2,b+1\} \ldots \) for \(b > 0\)

Examples:

\(0:\{1\}=1:\{0,0\}=2:\{0\}=3:\{\}=3\),so \(0:\{1\}=3\)

\(0:\{2\}=1:\{1,1\}=2:\{0,0,0,2\}=3:\{0,0,3\}=4:\{0,4\}=5:\{5\}=5^5=3125\),

so \(0:\{2\}=3125\)

\(0:\{3\}=1:\{2,2\}=2:\{1,1,1,3\}=3:\{0,0,0,0,1,1,4\}=4:\{0,0,0,1,1,5\}=5:\{0,0,1,1,6\}\)

\(=6:\{0,1,1,7\}=7:\{1,1,8\}=8:\{0^9,1,9\}=9:\{0^8,1,10\}\ldots =17:\{1,18\}=18:\{0^{19},19\}\)

\(=37:\{38\}=37^{38}\),so \(0:\{3\} = 37^{38}\approx 3.905 \times 10^{59}\)

\(0:\{4\}=1:\{3,3\}=2:\{2,2,2,4\}=3:\{1,1,1,1,2,2,5\}=4:\{0,0,0,0,0,1,1,1,2,2,6\}\)

\(=9:\{1,1,1,2,2,11\} =10:\{0^{11},1,1,2,2,12\}=21:\{1,1,2,2,23\}=22:\{0^{23},1,2,2,24\}\)

\(=45:\{1,2,2,47\}=46:\{0^{47},2,2,48\}\ldots=93:\{2,2,95\}=94:\{1^{95},96\}\ldots \)

Any idea for the growth rate?

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