Let us say a proper substring is any continuous section of an original string. For example cat is a substring of concatenate. We will say that a proper substring is a substring that is not equal to the original string. For example concatenate is a substring of concatenate but not a proper substring. (single character strings have no substrings)

We will now define a sequence using these terms. The nth term in this sequence will be the smallest number such that there is a proper substring of its binary representation that is not a substring of any earlier term in the sequence. The first term is 10.

As an exercise lets generate the first 5 terms. I will work in binary to make things easier.

The first term is 10 since 11, the next smallest number, has only one proper substring, 1 which is also a substring of 10, 11 is not in the sequence. 100 however does contain the proper substring 00 which is not a substring of 10 so 100 is our next term. Next is 101 which contains the unique proper substring 01 adding it to the sequence, then 110 contains the proper substring 11 which is new adding it to the sequence.

Now we have

10, 100, 101, 110

111 is up next but it contains only the substrings 1 and 11 making it not a term. 1000 however contains 000 adding it to the sequence.

Here are the first couple terms in decimal

2, 4, 5, 6, 8, 9, 10, 11, 14, 16, 17, 18, 19, 20, 21, 22, 23, 24, 26, 30, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 50, 54, 56, 58

Task

Either

• Take n as input and generate the nth term in this sequence (either 0 or 1 indexed)

• Continuously output terms of the sequence

This is code-golf answers are scored in bytes with less bytes being better.