Puzzle ZAHU 

Workings

Hint
The cube root of the given number is a good starting point.

Answer
71, 73, 79.

Reasoning
The cube root of 409,457 is around 74.

The closest consecutive prime numbers to 74 are: 61, 67, 71, 73, 79, 83, 89, 97.

Using a calculator, it is quickly seen that the answer can only be found using 71, 73 and 79.