For any positive integer n, let f(n)= log_8n, & log_8n∈Q, 0, & log_8nQ. What is ∑_n=1^1997f(n)?

1997 AMC 12 Problem 21

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21.

For any positive integer

n,

let

f(n)= \begin{cases} \log_8n, & \log_8n\in\mathbb{Q},\\ 0, & \log_8n\notin\mathbb{Q}. \end{cases}

What is

\displaystyle\sum_{n=1}^{1997}f(n)?

\log_8 2047

6

\frac{55}{3}

\frac{58}{3}

585

Determine which integers can be rational powers of

8=2^3

List the powers of

2

not exceeding

1997

and sum their base-

8

logarithms

For integer

n,

\log_8n

is rational exactly when

n

is a power of

2.

The relevant values are

n=2^k

for

0\le k\le10,

and

f(2^k)=\frac{k}{3}.

Therefore the sum is

\frac{0+1+\cdots+10}{3}=\frac{55}{3},

so C is correct.