Question

The sequence 2, 3, 5, 6, 7, 10, 11, $\ldots$ contains all the positive integers from least to greatest that are neither squares nor cubes. what is the $400^{\mathrm{th}}$ term of the sequence?

Answer 1

The 400th term is 425.

There are floor(√400) = 20 squares in the range 1..400, so the 400th term will be at least 420. There are floor(∛420) = 7 cubes in the range 1..400, so the 400th term may be as high as 427. However, there are $\lfloor\sqrt[6]{427}\rfloor=2$ numbers that are both squares and cubes. Consequently, the 400th term will be 427-2 = 425.