Complementary sequence of another sequence

Question

I have a infinite or finite sequence of positive integers. Let it be infinite for this example:

The infinite sequence $\{1,2,3,4,6,7,8,9,11,12,13,14,16,17,18,19,...\}$ shows that steps of $5$ are left out. So if my notation (on congruent relation) is valid, numbers that is $n\equiv 0 \mod 5$ are left out.

My question is simply if we include the numbers that where left out in the previous sequence, and put the missing ones into another sequence (creating a new sequence) like this: $\{5,10,15,20,25,...\}$, what is this last sequence called?

A complementary sequence? Or can we use the terms of set-theory, a complementary set? Im hoping to learn more terminologies of these concepts, so I just start by asking a simple question. How can I write the above with math notation?

I also ask for a good book (that is not too dry) on the subject without much advanced formulations.

Answer 1

"Complementary sequences" is the name of this phenomenon. You can read about this here:

https://www.fq.math.ca/Papers1/48-4/Mortici.pdf

After you understand the article, you can demonstrate easily that

$$NS(n)=n+1+\left\lfloor\cfrac{1+\sqrt{4n+1}}{2}\right\rfloor$$

misses exactly the squares!

Just out of curiousity, the sequence

$$Z(n)=n+1+\left\lfloor \cfrac{x}{N-1} \right\rfloor$$ misses exactly the multiples of $N$.

Answer 2

The second set is the relative complement of the first set with respect to the natural numbers. If the first set is $A$, then the complement of $A$ relative to the set $U$ is written as $B = A^c = U \backslash A$.