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f3(1)=f2(1)=2^1*1=2 an interesting note about 2 it is the only even prime number all the others are odd.

f3(2)=f2(f2(2)=f2(8)=2^8*(8)=2048 if you add the digits together you get a prime 2+0+4+8=2+12=2+1+2=3+2=5.

f3(3) also known as fw(3)= f2(f2(f2(3)=f2(f2(24)=f2(2^24*(24)=402653184 added together all the digits come to six which is the first perfect number. We can prove this very easily 4+0+2+6+5+3+1+8+4=4+8+8+9+4=4+16+13=4+29=33=3+3=6.

f3(4)=f2(f2(f2(f2(4)=f2(f2(f2(64)=f2(f2(1.180591620717 × 10^21)=f2( 2.676043612057 × 10^42)=1.710261754098 × 10^85.

f3(5)=f2(f2(f2(f2(f2(5)=f2(f2(f2(f2(160)=f2(f2(f2(2.338402619729 × 10^50)=f2(f2( 1.18262925860 × 10^101)=f2( 2.68445111670 × 10^202)=1.72566228676 × 10^405.

f3(6)=f2(f2(f2(f2(f2(f2(6)=f2(f2(f2(f2(f2(384)=f2(f2(f2(f2(1.51303703794 × 10^118)=f2(f2(f2(4.31836246445 × 10^236)=f2(f2(8.61540556184 × 10^473)=f2(3.37886799509 × 10^949)=3.5148908494×10^1899.

f3(7)=f2(f2(f2(f2(f2(f2(f2(7)=f2(f2(f2(f2(f2(f2(896)=f2(f2(f2(f2(f2(4.73351899898 × 10^272)=f2(f2(f2(f2(1.25925976797 × 10^546)=f2(f2(f2(3.0143264946 × 10^1092)=f2(f2(2.4355271903 × 10^2185)=f2(1.3175276007 × 10^4371)=3.2837875342 × 10^8742

f3(8)=f2(f2(f2(f2(f2(f2(f2(f2(8)=f2(f2(f2(f2(f2(f2(f2(2048)=f2(f2(f2(f2(f2(f2(6.61852284341 × 10^619)=f2(f2(f2(f2(f2(6.5033177646 × 10^1240)=f2(f2(f2(f2(5.8996832252 × 10^2482)=f2(f2(f2(3.5221699635 × 10^4966)=f2(f2(4.0465891991 × 10^9933)=f2( 6.687038861 × 10^19867)=6.890220596 × 10^639736.

As you can see f3(n) grows very rapidly but it is still painfully slow compared to f4(n) or f(100)n etc.

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