Does storing an apple with potatoes prevent the potatoes from sprouting?
According to this article, the advice to not store onions and potatoes together has nothing to do with ethylene. Onions do not give off ethylene. It has to do with moisture. Both onions and potatoes give off some moisture. Storing them together makes both more likely to rot.

Apples, on the other hand, do give off ethylene. This begs the question, &quot;What does ethylene gas do to potatoes in storage?&quot; Here's what I found.

Ethylene exerts a dual effect on potato tubers: it markedly shortens the duration of rest, but it inhibits elongation of the sprouts during extended treatment. Source


Ethylene has been reported to break endodormancy following short-term treatments, but also to inhibit sprout growth and promote ecodormancy when supplied continuously – either starting immediately after harvest or at first indication of sprouting. - Source


“Endodormancy” occurs after harvest and is due to the internal or physiological status of the tuber. In this situation, even if tubers are placed in conditions favorable for sprout development, sprouting will not occur. “Ecodormancy” is when sprouting is prevented or delayed by environmental conditions. An example of this would be potatoes stored at lower temperatures having a longer dormancy period compared to potatoes stored at warmer temperatures. Source


short-term exposure of potatoes to ethylene gas encouraged more sprouting while long term exposure suppressed it. Source.

Summary:
If your potatoes are in the endodormant stage ethylene will break them out of endodormancy and move them into the ecodormant stage. Potatoes in the endodormant stage are very shelf-stable. Potatoes in the ecodormant stage are less shelf stable.
Once your potatoes are already in the ecodormant stage, there's no disadvantage to supplying them with ethylene. Ethylene will inhibit sprout growth at this stage.
But, you can't tell if your potatoes are in the endodormant stage or the ecodormant stage. The only way you can tell what dormancy stage they're in, is when they start to sprout. Then you know they're not dormant anymore.
So, you want to either:

Supply your potatoes with ethylene continuously, or
Wait until they start to sprout, then supply them with ethylene.

Once you have supplied your potatoes with ethylene, don't take away the source of ethylene.

Conclusion:
Don't:

Put an apple in with your potatoes for a while, then take it away. The ethylene will &quot;wake up&quot; your potatoes, and they will sprout sooner than if you didn't put an apple with them in the first place.
Put an apple in with your potatoes and forget about it. If the apple goes bad and starts to rot, your potatoes will be more likely to rot.

Do:

Put an apple in with your potatoes as soon as you put them in storage, remembering to check on them and replacing the apple if it goes bad. Your potatoes will sprout slightly less than without the apple.

-Or-

Store your potatoes without an apple, but check on them occasionally for sprouting. When they start to sprout, put an apple in with them. The apple will slow the sprout growth and make your potatoes last a bit longer.

Both options require you to check on your potatoes occasionally. If you don't want to do this (or don't trust yourself to remember), don't store them with an apple. Better to risk a bit of sprouting than the rot that a rotting apple will cause.