Definition:Two-Sided Prime

Definition

A two-sided prime is a prime number which remains prime when:

any number of digits are removed from the left hand end

and:

any number of digits are removed from the right hand end

but, generally, not from both ends at once.

Sequence

The complete sequence of two-sided primes is:

$2, 3, 5, 7, 23, 37, 53, 73, 313, 317, 373, 797, 3137, 3797, 739 \, 397$

Examples

$739 \, 397$ is a Two-Sided Prime

 $\displaystyle 739 \, 397$  $\displaystyle$ is the $59 \, 489$th prime $\displaystyle 39 \, 397$  $\displaystyle$ is the $4148$th prime $\displaystyle 9397$  $\displaystyle$ is the $1162$nd prime $\displaystyle 397$  $\displaystyle$ is the $75$th prime $\displaystyle 97$  $\displaystyle$ is the $25$th prime $\displaystyle 7$  $\displaystyle$ is the $4$th prime

 $\displaystyle$  $\displaystyle 739 \, 397$ is the $59 \, 489$th prime $\displaystyle$  $\displaystyle 73 \, 939$ is the $7296$th prime $\displaystyle$  $\displaystyle 7393$ is the $939$th prime $\displaystyle$  $\displaystyle 739$ is the $131$st prime $\displaystyle$  $\displaystyle 73$ is the $21$st prime $\displaystyle$  $\displaystyle 7$ is the $4$th prime

$\blacksquare$