There are so many illion-related notations in googology, but it's time for me to make one. This is based on recursion and hyperrecursion and works a bit like the fast-growing hierarchy.
E(n) = 10^(3*10^(3*10^(...(3*10^n)...))+3) with n 3*10's
E0(n) = E(n)
Em(n) = Em-1(Em-1(...(Em-1(n))...)) with n sets of brackets
En(n) = E[1](n) = order type w
2-entry array notation
We can now set forth three rules, where @ is a string of numbers and separators. All numbers in all arrays using this notation must be nonzero positive integers.
Rule 1. Tailing rule: E[@,1](n) = E[@](n)
Rule 2. Recursion rule: E[m,k](n) = E[m-1,k](E[m-1,k](...(E[m-1,k](n))...)) with n nests
Rule 3. Hyperoperation rule: E[n,k](n) = E[1,k+1](n)
The limit of this notation is order type w^2.
linear array notation e.g. E[a,b,c,...,n](x)
Now we'll have to edit rule 2.
E[m@](n) = E[m-1@](E[m-1@](...(E[m-1@](n))...)) with n nests
If rule 1, 2, 3 or the rule above does not apply, we start a thing called process which starts from the first number after the opening bracket.
Case A. If the number is 1, jump to the next entry.
Case B. If the number is greater than 1, decrease the entry by 1 and change the previous entry to the number in the () bracket. Check if rules apply.
E.g. E[1,2,3](3) = E[3,1,3](3)
This brings the limit of the notation up to w^w.
My planned extensions to this notation has the array behave like the one in Username5243's Array Notation, except at epsilon-zero, the first-order comma becomes a double comma, a second-order comma becomes a triple comma, and the limit would be akin to order type psi_0(Omega_w).
The prefixes all follow Denis Maksudov's naming system.
E(0) = 1, because 10^0 = 1/zeralillion
E(1) = 10^33 (decillion)/unalillion
E(2) = hectillion/balillion
E(3) = kalillion/tralillion
E(4) = quadralillion
E(5) = quintalillion
E(9) = hairillion (nonalillion)
E(1000) = killalillion
E(10^6) = megalillion
E(10^21) = zettalillion
E(10^24) = yottalillion
E1(2) = E(E(2)) = hendunillion
E1(3) = E(E(E(3))) = hentrunillion
Rpillager's seven-e = E(E(E(10^6))) (the seven e stands for the original E(E(E(10^6))) by Harry Foundalis) (note that this is smaller than the original)
E1(10) = heckillion
E2(10) = deckillion
E3(10) = treckillion
E4(10) = quadrackillion
E5(10) = quintackillion
E10(10) = E[1](10) = hyprunillion = dekackillion
E1000(10) = killackillion
E10^6(10) = megackillion
E[1](meameamealokkapoowa) = meameamealokkapockillion
and so on.
MORE COMING SOON
Copyright 2023 Redstonepillager. Work is licensed under CC-BY-SA.