Question

The increasing sequence $2,$ $3,$ $5,$ $6,$ $7,$ $10,$ $11,$ $\ldots$ consists of all positive integers that are neither the square nor the cube of a positive integer. Find the $500$ th term of this sequence.

Answer

Answer: 528

Concepts:counting integers in a range inclusion-exclusion perfect square

Difficulty rating: 1800

Small Hint:

Count squares and cubes up to a candidate endpoint by inclusion-exclusion

Big Hint:

Numbers that are both squares and cubes are sixth powers

Solution:

Through $529,$ there are $\lfloor\sqrt{529}\rfloor=23$ squares, $8$ cubes, and $2$ sixth powers counted in both groups. Thus the number of allowed terms at most $529$ is $529-23-8+2=500.$ But $529=23^2$ is excluded, while $528$ is neither a square nor a cube. Therefore the $500$ th term is $528.$

1990 AIME Problem 1

Answer: 528

Difficulty rating: 1800

Small Hint:

Big Hint:

Solution:

Problem 1 in Other Years