The hard part isn't figuring out the pattern, it's figuring out how we are supposed to extend it to find integers four and five.
Edit: The title contains each of the vowels (a,e,i,o,u) exactly once. The 4 and 5 in my hint refer to the number of digits in the two other answers.
A: When spelled out in English, the numbers contain the 5 vowels (a, e, i, o, u, but not y) exactly once.
The other two numbers would be 6,010 (six thousand ten) and 10,006 (ten thousand six).
I discounted answers like 80,000 and 90,000 which also contain y and wouldn't preclude 26,000 as an answer.