Jelly, 13 meaningful characters, language postdates challenge
R µ ọḊ *@Ḋ ċ >2 µ Ðf
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All whitespace here is insignificant. I used it to show the structure of my answer, as the question asks.
Here's how it works:
R µ ọḊ *@Ḋ ċ >2 µ Ðf
R Ðf Find all numbers n from 1 to the input, such that:
µ µ (grouping marks, like {} in C)
Ḋ Ḋ Take the range from 2 to n
ọ Find the number of times each divides n
*@ Raise the range from 2 to n to these powers
ċ Count the number of times n appears
>2 and the result must be greater than 2
So for example, when testing n=256, we check the number of times each of the numbers from 2 to 256 divides into 256. The only numbers that divide more than once are 2 (which divides 8 times), 4 (which divides 4 times), 8 (which divides twice), and 16 (which divides twice). So when we raise the number of divisions to the powers determined there, we get:
2⁸, 3, 4⁴, 5, 6, 7, 8², 9, 10, 11, 12, 13, 14, 15, 16², 17, ..., 255, 256
This produces the original value, 256, a number of times equal to the way that 256 is a perfect power, plus one (the last element produces 256 because 256 = 256¹). So if we see 256 more than twice in the array (and we do in this case; 8² is 64 but the other "interesting" elements all produce 256), it must be a perfect power.