Graph traversal, maybe another type of mathematics?
If you really intended to find the minimum amount, the answer is 0, because you don't have to use any number at all.
I guess you meant to write "maximal amount of numbers". 
If I understand your problem correctly, it sounds like we can translated it to the following problem:
Given a set of n numbers (1,..,n), what is the maximal amount of numbers I can use to divide the set into pairs, where each number can appear only once. 
The answer to this question is:

when n = 2k    f(n) = 2k for k>=0
when n = 2k+1  f(n) = 2k for k>=0

I'll explain, using induction.

if n = 0 then we can use at most 0 numbers to create pairs.
if n = 2 (the set can be [1,2]) then we can use both numbers to
create one pair (1,2)
Assumption: if n=2k lets assume we can use all 2k numbers to create 2k pairs and prove using induction that we can use 2k+2 numbers for n = 2k+2.
Proof: if n = 2k+2, [1,2,..,k,..,2k,2k+1,2k+2], we can create k pairs using 2k numbers (from our assomption). without loss of generality, lets assume out pairs are (1,2),(3,4),..,(2k-1,2k). we can see that we still have two numbers [2k+1, 2k+2] that we didn't use, and therefor we can create a pair out of two of them, which means that we used 2k+2 numbers.

You can prove on your own the case when n is odd.
Let’s say you have a set/list/collection of numbers: [1,3,7,13,21,19] (the order does not matter). Let’s say for reasons that are not important, you run them through a function and receive the following pairs:
(1, 13), (1, 19), (1, 21), (3,19), (7, 3), (7,13), (7,19), (21, 13), (21,19). Again order does not matter. My question involves the next part: how do I find out the minimum amount of numbers that can be part of a pair without being repeated? For this particular sequence it is all six. For [1,4,2] the pairs are (1,4), (1,2), (2,4). In this case any one of the numbers could be excluded as they are all in pairs, but they each repeat, therefore it would be 2 (which 2 do not matter). 
At first glance this seems like a graph traversal problem - the numbers are nodes, the pairs edges. Is there some part of mathematics that deals with this? I have no problem writing up a traversal algorithm, I was just wondering if there was a solution with a lower time complexity. Thanks!