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## Higher-order legions Edit

These numbers can be defined or approximated only using higher-order analogues of legions, in which Bowers' dubbed them (from ascending order) lugions, lagions, ligions, etc. He also used the L notation to devise analogues of arbitrarily high order.

• Goshomity, {100,100 \\\\\\...\\\\\\ 2} (100 \'s)
• Good goshomity, {100,100 \\\\\\...\\\\\\ 2} (goshomity \'s)
• Great goshomity, {100,100 \\\\\\...\\\\\\ 2} (good goshomity \'s)
• Big bukuwaha, {L2,bukuwaha}100,100
• Bongo bukuwaha, {L3,big bukuwaha}100,100
• Quabinga bukuwaha, {L4,bongo bukuwaha}100,100
• Meameamealokkapoowa, {L100,10}10,10
• Meameamealokkabipoowa, {LL100,10}10,10
• Meameamealokkatripoowa, {LLL100,10}10,10
• Meameamealokkapoowa oompa, {LLL...LLL,10}10,10 (meameamealokkapoowa (array of) L's)

## large BEAF numbers Edit

They are the largest known computably large numbers devised. Numbers generated by Finite promise games and the Greedy clique sequence may also belong to this group.

But, yes. There is some huge n value at which point {3,n,1,2} starts to be at par with Loader's number as Loader's number is fixed but n can be close to infinity.

The logic is simple, it is sure that {3,Loader's number,1,2} >> Loader's number; so n must be far smaller than Loader's number. Range for n: SCG(13) < n < Loader's number

In 2013, the busy beaver number became computable, because n must be only a few millions for FOST(n) = busy beaver number. This means, it only takes a few million symbols, which can be easily stored by a computer (source: What are some of the values of Rayo(n)).

## computable numbers way beyond Rayo's number Edit

TREERayo's number(3) = TREE(TREE(...TREE(3)...)), where the total nesting depth of the formula is Rayo's number levels of the TREE function.

SSCGRayo's number(3) = SSCG(SSCG(...SSCG(3)...)), where the total nesting depth of the formula is Rayo's number levels of the SSCG function.

SCGRayo's number(13) = SCG(SCG(...SCG(13)...)), where the total nesting depth of the formula is Rayo's number levels of the SCG function.

DRayo's number(99) = D(D(D...D(D(99))...))), where the total nesting depth of the formula is Rayo's number levels of the D function.

TREE(n), SSCG(n), SCG(n) and D(n) are all 4 computable and with nestings this long, it is sure, that these 4 entries are larger than Rayo's number.

The TREE-nesting was originally used by Adam P. Goucher and Ying Zheng's claim, that the Rayo hierarchy is the only way to exceed Rayo's number is wrong. There is some huge d value at which point Dd(99) starts to be at par with Rayo's number, as Rayo's number is fixed as Rayo(10100), but d can be close to infinity. SCGc(13), SSCGb(3) and TREEa(3) can also be as big as Rayo's number, if c, b and a are large enough.

It is sure, that DRayo's number(99) >> SCGRayo's number(13) >> SSCGRayo's number(3) >> TREERayo's number(3) >> Rayo's number. So, for "Dd(99) = SCGc(13) = SSCGb(3) = TREEa(3) = Rayo's number", d, c, b and a must all 4 be far smaller than Rayo's number. Range for d, c, b and a: Loader's number < d < c < b < a < Rayo's number

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