#skewes

A129935 - Numbers n such that ceiling( 2/(2^{1/n}-1) ) is not equal to floor( 2n/(log 2) )

777451915729368, 140894092055857794, 1526223088619171207, ...

Something a little different today. The other day on Reddit's Math subsite, there was a thread about statements that appeared true for all n less than something large. The first example given was Skewes' Number, a number where it's been proven that the number of primes is finally greater than the logarithmic integral. The definition isn't important, but if you had tested the first hundred quintillion integers (which would take a while) you might have convinced yourself that π(n) < li(n) forever. But it's not. Since 1933, Skewes' number has shrunk from the original eee79 -- it is now somewhere around 10316.

One comment links to a question on Math Overflow with a similar premise. That thread links to several interesting integer sequences in addition to the above. For instance, A102567 are counterexamples to "There is no positive integer n such that the concatenation of n with itself is a square" -- the first is 13223140496.