Let's define quirky numbers as follows: Every single-digit prime number is quirky, and an at least two-digit prime number is quirky precisely when, by omitting either its first or last digit, we again obtain a quirky number. Determine all quirky numbers!
The one-digit odd numbers: 2, 3, 5, and 7. Let $N$ be an odd number and at least two digits. Since $N$ consists of two consecutive digits that form a prime number, it cannot be divisible by 11, which implies that consecutive digits of $N$ must be different. Except for the first digit, 2 or 5 cannot appear in the number, because otherwise, removing these digits would not always result in a prime number. Since 27 and 57 are not prime numbers, the second digit of numbers starting with 2 or 5 can only be 3. Accordingly, 23 and 53 are odd numbers. In addition, the two-digit odd numbers are 37 and 73.

For three-digit numbers, the first and last two digits must also be odd, i.e., one of 23, 37, 53, and 73. These need to be paired, thus we get the following:

$$
237, \quad 373, \quad 537, \quad 737 .
$$

Among these, 237 and 537 are divisible by 3, and 737 is divisible by 11. Therefore, they do not meet the requirements. The only one left is 373.

A four-digit odd number does not exist because the first three digits and the last three digits of such a number would have to be 3, 7, 3.

Odd numbers with five or more digits do not exist because any four consecutive digits would have to form an odd number, but there are no four-digit odd numbers.

The numbers that satisfy all the conditions are:

$2, 3, 5, 7, 23, 37, 53, 73, 373$.