Goodstein's sequence

g_b(n) can be constructed as follows:
First, write n in base b with all exponents (and exponents of exponents and so on) in base b.

eg. for n=529 and b=2, 529=2⁹+2⁴+1=2²³⁺¹+2²²+1=2²²⁺¹⁺¹+2²²+1

Now, replace each occurence of b by b+1 and finally substract 1 to get g_b(n)

eg. g₂(529)=3³³⁺¹⁺¹+3³³ (which is approximately 1.33x10³⁹)

the sequence g_b(n), g_b+1(g_b(n), g_b+2(g_b+1(g_b(n)),... is called the Goodstein sequence. As you might guess, this sequence grows really fast to really big numbers, however...