TITLE: Longest coinciding pair of integer sequences known
QUESTION [21 upvotes]: There are arbitrarily many pairs of integer sequences (of arbitrary origins) that coincide upto an $N$ but differ for an $n > N$. I assume, the coincidence will be considered accidentally then by default, but I may be mistaken about that.
One is disadviced to draw any conclusions from coincidences of integer sequences unless its proven, that they coincide for all $n$. (Even then there may be no sensible conclusions, as I have learned here: Equivalence of families of objects with the same counting function.)
In any case, it is hard not to be entrapped to draw a conclusion when $N$ is very large. But what is "very large"? Thus my question:

What is the largest $N$ with two known
  integer sequences coinciding upto $N$
  but differing for an $n > N$?

(Can this information be captured from OEIS by an intelligent query?)
(I am aware of the fact that one can trivially define pairs of integer sequences which conincide for all $n$ but a single and arbitrarily large one. It should be clear that I am not interested in those but in pairs that are not adjusted to each other this way.)

REPLY [9 votes]: A possibly unbeatable example: the sequence $a(n)$, where $a(n)$ is the periodicity of the first row in the Laver table based on the set $\left\lbrace 1,\ldots, 2^n\right\rbrace$:
The first values are $$1, 1, 2, 4, 4, 8, 8, 8, 8, 16, 16,\ldots$$ 
and then it stays at $16$ for ages:
Randall Dougherty showed that the first $n$ for which $a(n)$ can possibly differ from $16$ is A(9,A(8,A(8,255))), where A denotes the Ackermann function, a recursive function whose first values are already huge.
However, it can be shown that the sequence actually tends to infinity, under some additional axiom that is independent to the usual ones from ZFC.
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