Infinitely many cards

We see that 1 is written once, then 2 is written once, 3 is written two times and 4 is written 2 times etc.It looks as if the number n is written on $\varphi (n)$ cards.And indeed let it be true for all numbers less than n.Then the divisors of n would be written on

$\displaystyle\sum_{d|n,d<n}^{} \varphi (d)=\sum_{d|n}^{} \varphi(d) - \varphi(n)=n-\varphi(n)$ cards.We used the well-known identity

$\displaystyle\sum_{d|n}^{} \varphi(d)= n$ .

Then for n would be left

$n-(n-\varphi(n))=\varphi(n)$ cards.

What we are looking for is $\varphi(2^{10})=2^9=\boxed{512}$