Barrier

A number $n$ is called a barrier of a number-theoretic function $f(m)$ if, for all $m<n$, $m+f(m)\leq n$. Neither the Totient Function $\phi(n)$ nor the Divisor Function $\sigma(n)$ has a barrier.

References

Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 64-65, 1994.