Persistent Number

An $n$-persistent number is a Positive Integer $k$ which contains the digits 0, 1, ..., 9, and for which $2k$, ..., $nk$ also share this property. No $\infty$-persistent numbers exist. However, the number $k=1234567890$ is 2-persistent, since $2k=2469135780$ but $3k=3703703670$, and the number $k=526315789473684210$ is 18-persistent. There exists at least one $k$-persistent number for each Positive Integer $k$.

See also Additive Persistence, Multiplicative Persistence

References

Honsberger, R. More Mathematical Morsels. Washington, DC: Math. Assoc. Amer., pp. 15-18, 1991.