I was reading "Concrete Mathematics" and confronted the Joseph Problem. I'm astonished that the recurrence equation in binary representation is so simple. i.e. for J(n) say J(345), $\text{J(345) = J(}101011001_2\text{)=}10110011_2$
the recurrence function is to shift the binary representation left by one bit. While the representation in base 10 is more complicated:
$\text{f(2n) = 2 f(n)-1}$
$\text{f(2n+1) = 2 f(n)+1}$
So I'm wondering if representation in other bases(binary, etc.) may aid to the analyzation of number patterns, hence has some application in number theory.