The ultron is defined like so:[1]
- Define \(u_{0,1}\) as \(h_{100}(10,10,10,10,10,10,10,10,10,10)\), using the hyperlicious function.
- Define \(u_{x,1}\) as \(h_{100}(\underbrace{10,10,\ldots,10,10}_{u_{x - 1,1}})\).
- Define \(u_{0,2}\) as \(\underbrace{u_{u_{u_{\ddots_{100,1}},1},1}}_{10 \text{ copies of } u}\).
- Define \(u_{x,2}\) as \(\underbrace{u_{u_{u_{\ddots_{100,1}},1},1}}_{u_{x - 1,2} \text{ copies of } u}\).
- Define \(u_{0,3}\) as \(\underbrace{u_{u_{u_{\ddots_{100,2}},2},2}}_{10 \text{ copies of } u}\).
- Define \(u_{x,3}\) as \(\underbrace{u_{u_{u_{\ddots_{100,2}},2},2}}_{u_{x - 1,3} \text{ copies of } u}\).
- Continuing this process, the ultron is \(u_{100,100}\).
The ultron is comparable to \(f_{\omega+200}(100)\) in the fast growing hierarchy.
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Numbers from A googol is a tiny dot
Hypermathematics: bigoogol · trigoogol · quadrigoogol · coogol(plex)
Hyperlicious: wakoogol(plex) · wakamoogol(plex) · wonkapoogol(plex) · ultron
Numbers with a W: woogol · wiggol · waggol · weegol · wigol · woggol · wagol · bwoogol · bwiggol · bwaggol · bweegol · bwigol · bwoggol · bwagol
Primes: Gooprol · Booprol · Trooprol · Quadrooprol
Other: Bentley's Number · Pigol · Egol · Phigol · gongol(plex) · kaboodol(plex) · gaz(illion)