Is there a base $b\geq11$ that we obtain a perfect power after we concatenate the first $n$ positive integers of that base?
In base ten, the concatenation of first $n$ positive integers does not give any other perfect power other than $1$.
But what about on other bases?
Note that I excluding cases like $1, 12, 123$, as those are somewhat trival.
I did already check all bases greater than ten, but $\leq36$ to check if there is a base $b$ such that $1234_{b}, 12345_{b}$ and $123456_{b}$ are perfect powers, but there is none.
I mean when I say concatenation on other bases, it is like this, example in base twenty-two:
$1_{22},12_{22},123_{22},\cdots,123456789abcdefghjkl10_{22}$.
In general, $1,12,123,\cdots,123\cdots(b-1)10$.
I know these things:
$6$ is never the last digit of a perfect power on bases that are divisible by four or nine.
$2$ is never a perfect power on bases that is divisible by four.
By using Phyton, I tried to check all bases from eleven to thirty-six to check if are there any perfect powers that is obtained by concatenating the first $n$ positive integers of that base, but it was unsuccessful.