I've noticed that people usually say at other times.
However, I've also seen instances of other times, without
the preposition at.
When do we use at and when don't we need it when we
use other times?
Other times can be used with or without the preposition
at, often interchangeably. The phrase needs to fit
comfortably in the sentence. Of course, there will be a reference to
an earlier event. For example:
Sometimes he would arrive late; other times he would
not arrive at all.
At certain times he would arrive late; at other times he
would not arrive at all.
These two sentences are fine the way they are. The at
or absence of at is parallel with the construction
in the first clause. Or you could add at to the first
sentence and omit it in the second sentence.
Other times seems a bit more descriptive of the event,
while at other times focuses on noting the event in
a time frame. You can get your clue for which phrase works better by
connecting it to the previous reference.