User blog:Googleaarex/Hardy

From 0 to \(\omega^\omega)\
\(H_0(n) = n\)

\(H_1(n) = n+1\)

\(H_2(n) = n+2\)

\(H_m(n) = n+m\)

\(H_\omega(n) = 2n\)

\(H_{\omega+1}(n) = 2(n+1) = 2n+2\)

\(H_{\omega+2}(n) = 2(n+2)\)

\(H_{\omega+m}(n) = 2(n+m)\)

\(H_{\omega 2}(n) = 4n\)

\(H_{(\omega 2)+m}(n) = 4(n+m)\)

\(H_{\omega 3}(n) = 8n\)

\(H_{(\omega 3)+m}(n) = 8(n+m)\)

\(H_{\omega m}(n) = 2^n\)

\(H_{\omega m+x}(n) = 2^m(n+x)\)

\(H_{\omega^2}(n) = 2^n*n\)

\(H_{\omega^2+1}(n) = 2^{n+1}*(n+1)\)

\(H_{\omega^2+2}(n) = 2^{n+2}*(n+2)\)

\(H_{\omega^2+\omega}(n) = 2^{2n}*(2n)\)

\(H_{\omega^2+\omega 2}(n) = 2^{4n}*(4n)\)

\(H_{(\omega^2) 2}(n) = 2^{2^n*n}*(2^n*n)\)

\(H_{(\omega^2) 3}(n) = 2^{2^{2^n*n}*(2^n*n)}*(2^{2^n*n}*(2^n*n))\)

\(H_{(\omega^2) m}(n) = 2^{H_{(\omega^2) m-1}(n)}*(H_{(\omega^2) m-1}(n))\)

\(H_{\omega^3}(n) \approx n \uparrow\uparrow n\)

\(H_{(\omega^3) 2}(n) \approx n \uparrow\uparrow (n \uparrow\uparrow n)\)

\(H_{\omega^4}(n) \approx n \uparrow\uparrow\uparrow n\)

\(H_{\omega^5}(n) \approx n \uparrow\uparrow\uparrow\uparrow n\)

\(H_{\omega^m}(n) \approx \{n,n,m\}\)

Beyond \(\omega^\omega)\
\(H_{\omega^\omega}(n) \approx \{n,n,n\}\)

\(H_{\omega^\omega+1}(n) \approx \{n,n,n+1\}\)

\(H_{\omega^\omega+2}(n) \approx \{n,n,n+2\}\)

\(H_{\omega^\omega+\omega}(n) \approx \{n,n,2n\}\)

\(H_{\omega^\omega+\omega^2}(n) \approx \{n,n,2^n*n\}\)

\(H_{(\omega^\omega) 2}(n) \approx \{n,n,\{n,n,n\}\}\)

\(H_{\omega^{\omega+1}}(n) \approx \{n,n,1,2\}\)

\(H_{\omega^{\omega+2}}(n) \approx \{n,n,2,2\}\)

\(H_{\omega^{\omega+m}}(n) \approx \{n,n,m,2\}\)

\(H_{\omega^{\omega 2}}(n) \approx \{n,n,n,2\}\)

\(H_{\omega^{\omega 2+1}}(n) \approx \{n,n,1,3\}\)

\(H_{\omega^{\omega 2+2}}(n) \approx \{n,n,2,3\}\)

\(H_{\omega^{\omega m+2}}(n) \approx \{n,n,m,3\}\)

\(H_{\omega^{\omega m}}(n) \approx \{n,n,n,m\}\)

\(H_{\omega^{\omega m+x}}(n) \approx \{n,n,x,m+1\}\)

\(H_{\omega^{\omega^2}}(n) \approx \{n,n,n,n\}\)

\(H_{\omega^{\omega^3}}(n) \approx \{n,n,n,n,n\}\)

\(H_{\omega^{\omega^m}}(n) \approx \{n,m+2 (1) 2\}\)

\(H_{\omega^{\omega^\omega}}(n) \approx \{n,n+2 (1) 2\}\)

\(H_{\omega^{\omega^\omega}+1}(n) \approx \{n,n+3 (1) 2\}\)

\(H_{\omega^{\omega^\omega}+m}(n) \approx \{n,n+(m+2) (1) 2\}\)

\(H_{\omega^{\omega^\omega}+\omega}(n) \approx \{n,2n (1) 2\}\)

\(H_{\omega^{\omega^\omega}+((\omega) 2)}(n) \approx \{n,3n (1) 2\}\)

\(H_{\omega^{\omega^\omega}+\omega^2}(n) \approx \{n,n^2 (1) 2\}\)

\(H_{\omega^{\omega^\omega}+\omega^3}(n) \approx \{n,n^3 (1) 2\}\)

\(H_{\omega^{\omega^\omega}+\omega^\omega}(n) \approx \{n,n^n (1) 2\}\)