###Feel free to share this solution.###

#a(1)=2
#a(2)=3
#a(3)=5
#a(4)=7
#a(5)=11

#a(i) generates the i-th prime number.  So the 2 problems you have asked us to solve are famous open problems.  I will let you know when I have a solution.
#Happy April 1st!