Divide by the largest prime factor: ^(x+?)(?=\1+$)(?!(\1+)(\1\2)+$)

Of course, only useful if you don't need to capture the largest prime factor. It's in the same vein as the implicit division-by-odd-number done in (?!(x*)(\1\1)+$), which blew my mind. I went for 5 years thinking there were exactly two 17 character power-of-two regexes, and there were actually three.

@Grimy Regexes I was able to shorten using your division golf trick: Is this a consecutive-prime/constant-exponent number?, Is it a Proth number?, OEIS A033286 (prime × index),

and Is this number a factorial?

Ooh

Some of these I hadn’t seen before

Doesn’t the largest prime factor trick get some use as well?

Additionally, the OEIS A033286 one was able to benefit from your shorter largest-prime-factor

Another equal-length (30 char) form for "divide by the largest prime factor": (x+?)(?=(\1+?)(\1\2)*$)(?=\2$)

The molecular lookahead version is also the same length: (?*(x+?)\1+$)(?!(\1\1+)\2+$)\1

40 on Proth

@Grimy How?

Oh, I might see

It will be harder to comment

Nope, I don't see

Oh I think I do see

No, I don't see.

Interesting back and forth x)

@Grimy Did you intentionally leave a bit in your regex unoptimized so I could have the pleasure of finding it? :)

I didn’t

Oooh \2 as a shortcut for (xx)

Great find

:D

@Grimy You didn't update the comments. I'd put in the bit that \2 = 2