Is there any perfect power in the sequence $12,123,1234,12345,...$?

Question

Inspired from the question Is there any perfect square in the sequence $12,123,1234,12345,...$?, there is no perfect square other than $1$ in the sequence of Smarandache numbers. But I wonder if are there any perfect powers other than $1$ in the sequence of Smarandache numbers?

Using Pari GP, I checked up to the 20th term of Smarandache numbers to see if are there any perfect powers in the sequence, but I wasn't succesful.

Are there any perfect powers other than $1$ in the sequence of Smarandache numbers?

Answer 1

Here is what I discovered so far about the existence of perfect powers of degree $> 2$ among the terms of the OEIS sequence $A352991$ (which includes all the terms of the sequence $A001292$): https://arxiv.org/pdf/2205.10163.pdf

I also agree with the old Kashihara's conjecture that the sequence $A001292$ does not contain any nontrivial perfect power.

Answer 2

A perfect power $x$ must not contain a prime divisor $p$ such that $p^2$ divides $x$. So some of the non-perfect squares are $4k+2,25k+\{5;10;15;20\}$ and $9k+\{3;6\}$

With the first two cases, we can conclude $a_n$ isn't a perfect power if $n=4k+2$ or $n=25k+\{5;10;15;20\}$. With the third one, by caculating the sum of $a_n$'s digits through $n$ we can say $a_n$ isn't a perfect power if $n=9k+\{2;3;5;6\}$