These numbers have the property that if you double them you get a perfect square
2, 8, 18, 32, 50, 72, 98, ...
And these have the property that if you multiply them by 3 you get a perfect cube
9, 72, 243, 576, 1125, ...
Some numbers are in both lists; the smallest is 72 and
2 x 72 = 122, 3 x 72 = 63.
It's not possible to extend this and find a number n such that 2n is a square, 3n a cube and 4n a fourth power. However it was asked on Quora what the smallest number is such that n such that 2n is a square, 3n a cube and 5n a fifth power. Alon Amit showed the answer is 215320524. In full this is
I decided to extend this to the next prime number, 7, and found the smallest n such that 2n is a square, 3n a cube, 5n a fifth power and 7n a seventh power is 21053140584790. This impressive number evaluates to
which has 234 digits.
I think we can continue and add 11, 13 etc. but at this point I ran out of steam!