User blog:Clarrity/FGH numbers

Here is a list of small FGH numbers, based on \(f_2(n) = n*2^n\):

\(f_2(1) = 1*2^1 = 2\)

\(f_2(2) = 2*2^2 = 8\)

\(f_2(3) = 3*2^3 = 24\)

\(f_2(4) = 4*2^4 = 64\)

\(f_2(5) = 5*2^5 = 160\)

\(f_2(6) = 6*2^6 = 384\)

\(f_2(7) = 7*2^7 = 896\)

\(f_2(8) = 8*2^8 = 2048\)

\(f_2(9) = 9*2^9 = 4608\)

\(f_2(10) = 10*2^10 = 10240\)

Now \(f_2(f_2(n)) = f_2(n*2^n) = n*2^n*2^{n*2^n} = n*2^{n*2^n+n}\)

\(f_2(f_2(1)) = 1*2^{1*2^1+1} = 2^3 = 8\)

\(f_2(f_2(2)) = 2*2^{2*2^2+2} = 2^{2*2^2+3} = 2^11 = 2048\)

\(f_2(f_2(3)) = 3*2^{3*2^3+3} = 3*2^{27} = 402653184\)

\(f_2(f_2(4)) = 4*2^{4*2^4+4} = 2^{4*2^4+6} = 2^{2^6+6} = 2^{70} \approx 10^21 = \text{sextillion}\)

\(f_2(f_2(5)) = 5*2^{5*2^5+5} = 5*2^{165} \approx 10^50 \approx \text{Number of atoms in Earth}\)

Fun fact: \(f_2(f_2(n))\) is a power of 2 for n = \(2^m\).