Prime, followed by the cube of a prime, followed by the square of a prime. Other examples? The numbers 7, 8, 9, apart from being part of a really lame math joke, also have a unique property. Consecutively, they are a prime number, followed by the cube of a prime, followed by the square of a prime. Firstly, does this occurence happen with any other triplet of consecutive numbers? More importantly, is there anyway to predict or determine when and where these phenomena will occur, or do we just discover them as we go?
 A: Catalan's conjecture implies that a square and a cube cannot be consecutive unless they are (0,1) or (8,9). So this is the only triplet satisfying this property.
A: Suppose the first prime is $x$ and $x+1=y^3$ where $y$ is also a prime. Then $y$ has be an odd number which implies $x$ is even. So $x$ cannot be a prime. Contradiction.
A: Of course this exact pattern cannot recur as Patrick Li easily observed, but there are plenty of neat coincidences to be found in any group of small numbers.  My favorite: of 243, 343, 443, the first is the fifth power of a prime, the second is the cube of a prime, and the third is a prime.
A: If you'll settle for a prime, cube of a prime, square of a prime in arithmetic progression (instead of consecutive), you've got $$5,27=3^3,49=7^2\qquad \rm{(common\  difference\  22)}$$ and $$157,\ 343=7^3,\ 529=23^2 \qquad \rm{(common\  difference\  186)}$$ and, no doubt, many more where those came from. A bit more exotic is the arithmetic progression $81,\ 125,\ 169$ with common difference 44: the 4th power of the prime 3, the cube of the prime 5, the square of the prime 13. So $$3^4,5^3,13^2$$ is an arithmetic progression of powers of primes, and the exponents are also in arithmetic progression. 
