  The starting point of our paper is Kashihara's open problem number $30$,
concerning the sequence $A001292$ of the OEIS, asking how many terms are powers
of integers. We confirm his last conjecture up to the $100128$-th term and
provide a general theorem that rules out $4/9$ of the candidates. Moreover, we
formulate a new, provocative, conjecture involving the OEIS sequence $A352991$
(which includes all the terms of $A001292$). Our risky conjecture states that
all the perfect powers belonging to the sequence $A352991$ are perfect squares
and they cannot be written as higher order perfect powers if the given term of
$A352991$ is not equal to one. This challenging conjecture has been checked for
any integer smaller than $10111121314151617181920212223456789$ and no
counterexample has been found so far.
