Summation of values of function.

Question

A function g(n) having whole numbers as its domain is defined as $g(n)=\log_8n$ ;$f(n)=g(n)$ if $g(n)$ is rational and $f(n)=O$ if $g(n)$ is irrational. What is $\sum_{r=1}^{2007}f(n)$ I figured out that to be rational n must take rational powers of 8, but then the rational numbers between any two numbers are infinite, so how do we sum them.

Answer 1

It is fairly easy to show that $g(n)$ is rational if and only if $n$ is a power of $2$ (*). Since $2^0=1$ and $2^{10}=1024<2047$ while $2^{11}=2048>2047$, we have

$$\sum_{r=1}^{2007}f(n)=\sum_{k=0}^{10}{\frac{k}{3}}=\frac{10\times11}{2\times3}=\frac{55}{3}$$

This is because:

$$g(2^k)=\log_8{2^k}=\frac{1}{3}\log_2{2^k}=\frac{k}{3}$$

and (*) is true because if $\log_8{n}=\frac{p}{q}$ then we have

$$8^{p/q}=n \iff 8^p=n^q \iff 2^{3p}=n^q$$ which holds iff $n$ is a power of $2$.