# Derek's sequence

Discrete Mathematics Level pending

Consider the sequence of complex numbers defined by the recurrence $a_{k+1} = \frac{a_k - i}{a_k + i} \text{ for } k \geq 1,$ and let $$a_1$$ be any complex number such that $$a_n$$ is defined for all positive integers $$n$$. For each $$a_1$$, let $$S(a_1)$$ be the set of all possible values of $$\displaystyle \sum_{k=1}^{p \text{ prime} } (-1)^ka_k$$, where $$p$$ ranges over the prime numbers. What is $\max_{a} |S(a)| ?$

This problem is posed by Derek K.

Details and assumptions

In this problem, $$i$$ is the imaginary unit, satisfying $$i^2=-1$$.

$$| S |$$ denotes the number of distinct elements of set $$S$$.

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