Your number is, in fact, not bigger!

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username5243
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Re: Your number is, in fact, not bigger!

Postby username5243 » Tue Nov 24, 2015 8:49 pm UTC

Continuing, we have:

n|((())) = n|(()()...()()) ~ f_ω^ω(n) ~ n{1[2]2}n
n|()((())) = n|((()))|((()))|...|((())) ~ f_ω^ω+1(n) ~ n{2[2]2}n
n|(())((())) = n|()()...()()((())) ~ f_ω^ω+ω(n) ~ n{1,2[2]2}n
n|(())(())(()()) = n|()()...()(())((())) ~ f_ω^ω+ω*2(n) ~ n{1,3[2]2}n
n|(()())((())) = n|(())(())...(())(()()) ~ f_ω^ω+ω^2(n) ~ n{1,1,2[2]2}n
n|(()()())((())) = n|(()())...(()())((())) ~ f_ω^ω+ω^3(n) ~ n{1,1,1,2[2]2}
n|((()))((())) = n|(()()...()())((())) ~ f_ω^ω*2(n) ~ n{1[2]3}n
n|((()))((()))((())) = n|(()()...())((()))((())) ~ f_ω^ω*3(n) ~ n{1[2]4}n
n|(()(())) = n|((()))((()))...((())) ~ f_ω^(ω+1)(n) ~ n{1[2]1,2}n
n|((()))(()(())) = n|(()()...()())(()(())) ~ f_ω^(ω+1)+ω^ω(n) ~ n{1[2]2,2}n
n|(()(()))(()(())) = n|((()))((()))...((()))(()()()) ~ f_ω^(ω+1)*2(n) ~ n{1[2]1,3}n
n|(()()(())) = n|(()(()))(()(()))...(()(())) ~ f_ω^(ω+2)(n) ~ f_ω^(ω+2)(n)
n|(()()()(())) = n|(()()(()))...(()()(())) ~ f_ω^(ω+3)(n) = n{1[2]1,1,1,2}n
...
n|((())(())) = n|(()()...()()(())) ~ f_ω^(ω*2)(n) ~ n{1[2]1[2]2}n
n|((()))((())(())) = n|(()()...()())((())(())) ~ f_ω^(ω*2)+ω^ω(n) ~ n{1[2]2[2]2}n
n|((())(()))((())(())) = n|(()()...()()(()))((())(())) ~ f_ω^(ω*2)*2(n) ~ n{1[2]1[2]3}n
n|(()(())(())) = n|((())(()))...((())(())) ~ f_ω^(ω*2+1)(n) ~ n{1[2]1[2]1,2}n
n|((())(())(())) = n|(()()...()()(())(())) ~ f_ω^(ω*3)(n) ~ n{1[2]1[2]1[2]2}n
n|((())(())(())(())) = n|(()()...()()(())(())(())) ~ f_ω^(ω*4)(n) ~ n{1[2]1[2]1[2]1[2]2}n
...
n|((()())) = n|((())(())...(())(())) ~ f_ω^ω^2(n) ~ n{1[3]2}n

And sending as a number (certainly not the highest!):

100|((()())) ~ f_ω^ω^2(100)
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