Rules for a(n):

No number appears more than once in the sequence.

The first term (0) is divisible by 1

Each additional term is the smallest positive integer available that makes the sum of the first n terms divisible by n.

Create a new sequence b(n) by the same rules but beginning with b(1)=1.

[1] Prove b(n)=a(n)+1 for all n.

[2] Prove if b(n)=m then b(m)=n