
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Given a prime  $p$  and positive integer  $k$ , an integer  $n$  with  $0 \le n < p$  is called a  $(p, k)$ -Hofstadterian residue if there exists an infinite sequence of integers  $n_0, n_1, n_2, \ldots$  such that  $n_0 \equiv n$  and  $n_{i + 1}^k \equiv n_i \pmod{p}$  for all integers  $i \ge 0$ . If  $f(p, k)$  is the number of  $(p, k)$ -Hofstadterian residues, then compute  $\displaystyle \sum_{k = 1}^{2016} f(2017, k)$ .

*Proposed by Ashwin Sah*

Present the answer in LaTex format: \boxed{Your answer}
user