The series which forms the basis of all the other series is:- $1,2,4,8,16,18$. Some other combinations are:- $1,2,3,7,14,22 ; 1,2,4,7,15,20 ; 1,2,4,8,13,21$. However, I obtained the basic combination by the following method:-
Step 1:- You definitely need "1".
Step 2:- You need two and also 3. So, the next numbers is 2.
Step 3:- Now you have $1,24$. So, the next numbers is 4.
Step 4:- Now you have $1,2,4$. So, the next numbers is 8
Step 5:- Now you have $1,2,4,8$. So, the next numbers is 16
Step 6:- Now you have $1,2,4,8,16$. So, the next numbers is 18 So the series is $1,2,4,8,16,18$.
My Question:- I don't know how to prove that there are more ( or exactly these many ) rigorously ( and why ). Any help would be appreciated.