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The least positive integer for which the equality $$ \left\lceil \frac{2}{2^{1/n}-1}\right\rceil = \left\lfloor
\frac{2n}{\log 2} \right\rfloor. $$ fails is $n=777,451,915,729,368$. See http://research.att.com/~njas/sequences/A129935http://oeis.org/A129935.

Another example that I like is the number $f(n)$ of inequivalent differentiable structures on $\mathbb{R}^n$. We have $f(n)=1$ if $n\neq 4$, while $f(4)=c$, the cardinality of the continuum.
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The least positive integer for which the equality $$ \left\lceil \frac{2}{2^{1/n}-1}\right\rceil = \left\lfloor
\frac{2n}{\log 2} \right\rfloor. $$ fails is $n=777,451,915,729,368$. See http://research.att.com/~njas/sequences/A129935.

Another example that I like is the number $f(n)$ of inequivalent differentiable structures on $\mathbb{R}^n$. We have $f(n)=1$ if $n\neq 4$, while $f(4)=c$, the cardinality of the continuum.