Smallest positive integer which is coprime to the last two predecessors and has not yet appeared; a(1)=1, a(2)=2
Python 3.5, 160 141 126 124 121 109 bytes



This is a simple implementation of the sequence's definition. Golfing suggestions welcome.

Edit: -17 bytes thanks to Leaky Nun. -9 bytes thanks to Peter Taylor. -6 bytes thanks to Sp3000 and switching to Python 3.5. 

import math;f=lambda n,r=[2,1],c=3:n&lt;2and r[1]or(c in r)+math.gcd(c,r[0]*r[1])&lt;2and f(n-1,[c]+r)or f(n,r,c+1)


Ungolfing:

import math
def f(n, r=[2,1], c=3):
    if n&lt;2:
        return r[1]
    elif (c in r) + math.gcd(c,r[0]*r[1]) &lt; 2:
        return f(n-1, [c]+r)
    else:
        return f(n, r, c+1)