Elements of Geometric Sequence from One where First Element is not Power of Number

Theorem
Let $G_n = \left\langle{a_n}\right\rangle_{1 \mathop \le i \mathop \le n}$ be a geometric progression of integers.

Let $a_1 = 1$.

Let $k \in \Z_{> 1}$.

Let $a_2$ not be a power of $k$.

Then $a_m$ is not a power of $k$ except for:
 * $\forall m, k \in \left\{{2, \ldots, n}\right\}: k \mathop \backslash \left({m - 1}\right)$

where $\backslash$ denotes divisibility.

Proof
By Form of Geometric Progression of Integers from One, the general term of $G_n$ can be expressed as:
 * $a_j = q^{j - 1}$

for some $q \in \Z$.

Let $k \nmid \left({m - 1}\right)$.

Then by the Division Theorem there exists a unique $q \in \Z$ such that:
 * $m - 1 = k q + b$

for some $b$ such that $0 < b < k$.

Thus:
 * $a_m = a^{k q} a^b$

which is not a power of $k$.