TITLE: Least positive integer $n$ such that the digit string of $2^n$ ends on the digit string of $n$
QUESTION [9 upvotes]: What is the least positive integer $n$ such that the digit string of $2^n$ ends on the digit string of $n$:
  $$ (2^n)_{10} = d_m \, d_{m-1} \cdots
 d_{q+1} \, (n)_{10} \\ (n)_{10} = d'_{q} \cdots d_1' \\ d_i, d'_j \in
 \{0, \ldots, 9 \} $$

As in $2^3$ would somehow end in 3, or $2^5$ would end in 5.
Frankly I don't even know where to start. Thanks in advance.

REPLY [2 votes]: This OEIS sequence is just what you are looking for: https://oeis.org/A064541. Least number is 36.