Answer
The frequency is independent of the amplitude $A$ and the frequency is proportional to $\sqrt{\frac{k}{m}}$
Work Step by Step
Frequency is measured in Hertz which is $s^{-1}$ in SI units
We can assume that $f = C~A^a~k^b~m^c$
$C$ is a dimensionless constant
$A$ is amplitude in units of $m$
$k$ is spring constant in units of $N/m = kg~s^{-2}$
$m$ is mass in units of $kg$
We can equate the units of the terms:
$m^a~kg^b~(s^{-2})^b~kg^c = s^{-1}$
We can consider the units $s$:
$-2b = -1$
$b = \frac{1}{2}$
We can consider the units $kg$:
$b+c = 0$
$c = -b$
$c = -\frac{1}{2}$
We can consider the units $m$:
$a = 0$
Then:
$f = C~A^0~k^{1/2}~m^{-1/2}$
$f = C~\sqrt{\frac{k}{m}}$
Therefore, the frequency is independent of the amplitude $A$ and the frequency is proportional to $\sqrt{\frac{k}{m}}$
