The primes $73$, $12763$ and $255127$, formed by the concatenation of two consecutive Mersenne numbers, have this property:
$73$ divides $10^\frac{73-1}{3}-1$
$12763$ divides $10^\frac{12763-1}{3}-1$
$255127$ divides $10^\frac{255127-1}{3}-1$
I wonder if there are other primes p which divide $10^\frac{p-1}{3}-1$ and in particular if there are other primes formed by the concatenation of two consecutive Mersenne numbers with that property.