Remove every N-th N

awk, 10 bytes

Input is expected on STDIN, one number per line.

++a[$1]%$1

Explanation

Keeps a counter for each number in an associative array, prints only if the counter value modulo n is not zero. Printing is implicit. Long version:

++a[$1]%$1{print $0}

Pyth, 18 15 14 10 9 bytes

f%/aYTTTQ

I think this is the first code I've written that has five consecutive variable references in nine bytes.

I wish the array manipulation solution (u.DG%HgxHGH{QQ, 14 bytes) weren't so long.

f%/aYTTTQ       Implicit: Q=input
                 lambda T:
    Y              Variable: starts as empty list.
   a T             Append T to Y. Mutates Y.
  /   T           Number of elts of Y that equal T.
 %     T         Modulo by T
f       Q       Filter that lambda over Q.

Try it here.

Jelly, 10 8 bytes

=;\S€%af

Thanks to @Sp3000 for golfing off 2 bytes!

Try it online!

How it works

=;\S€%af    Main link. Input: A (list)

 ;\         Compute the cumulative concatenations of the elements of A, i.e., the
            list of A's prefixes.
=           Compare the elements of the nth prefix with the nth element of A.
   S€       Compute the sum of the resulting arrays.
     %      Take the nth sum modulo the nth element of A.
      a     Logical AND; replace the nth remainder with the nth element of A
            if the remainder is non-zero.
        f   Filter; remove the zeroes (which do not occur in A) from the result.