I claim 103138 divides that number for all positive integer values of n.
This is done by observing that $\displaystyle a-b$ divides $\displaystyle a^n - b^n$ and then rearranging the given number into sums of such differences.
For example,
$\displaystyle 278 = 2141-1863$ and $\displaystyle -278 = 1492 - 1770$
Thus $\displaystyle 278|2141^n -1863^n + 1492^n - 1770^n$
Similarly $\displaystyle 371 = 2141 - 1770$ and $\displaystyle -371 = 1492 - 1863$
Thus $\displaystyle 371|2141^n -1863^n + 1492^n - 1770^n$
Since (371,278) = 1
$\displaystyle (371)(278) = 103138|2141^n -1863^n + 1492^n - 1770^n$