User blog:QuasarBooster/Potentially googological function from number theory?

Has anyone here watched the Numberphile video about 82000 or heard about the sequence (2,3,4,82000)? Each member, a(n), is the smallest integer greater than 1 that can be written with only 1s and 0s in bases 2 through n. No one has found a(6) and it's even conjectured not to exist. Apparently the search has gone up to around 10^10^7.

Naturally, my first instinct was that this problem looks very similar to other problems in googology. Has anyone else looked into this? If someday this sequence is proved not to end then I think we might have something decent on our hands! And if it really is total, I would be very curious about how quickly it grows.

What do you guys think? Also, I've written an obfuscated algorithm to generate this sequence not very surprising coming from me  for anyone who's interested. def a(n): def p(x,b): B=1 while b*B<=x:B*=b if B<2:return x<2 return x//B<2 and p(x%B,b) N=2 while 0 not in [p(N,i) for i in range(2,n+1)]:N+=1 return N