Integer issue

Problem: We write all the integers from 1 to 512 inclusive, and cross out some of these so that in the remaining list, no number is double of any other. What is the maximum number of integers in remaining list.

Approach: If we arrange 512 numbers in some sets containing doubling multiples of each number. And put them in columns and then count them. If this count is F(n) and C(k) be the number of integers in k th column which are less equals to 512. Then we can describe F(n) as sum of C(k) . Now question is what is the last non zero term in these finite sum here means the value of k here for last C(k).

Clarification: let column 0 contains 1,3,5.... Column 1 will contain next possible multiples of each 1,3,5...to maximize the subset. And so on next columns.