The [i]subnumbers[/i] of an integer $n$ are the numbers that can be formed by using a contiguous subsequence of the digits. For example, the subnumbers of 135 are 1, 3, 5, 13, 35, and 135. Compute the number of primes less than 1,000,000,000 that have no non-prime subnumbers. One such number is 37, because 3, 7, and 37 are prime, but 135 is not one, because the subnumbers 1, 35, and 135 are not prime.

[i]Proposed by Lewis Chen[/i]
1. **Identify the possible starting digits:**
   - A number must begin with one of the prime digits: 2, 3, 5, or 7.

2. **Determine the valid sequences for each starting digit:**
   - **Starting with 2:**
     - The next digit must be 3 (or no next digit).
     - Valid numbers: 2, 23.
     - Check if 237 is prime: 237 is not prime (divisible by 3).
     - Therefore, valid numbers are 2 and 23.

   - **Starting with 3:**
     - The next digit must be 7 (or no next digit).
     - Valid numbers: 3, 37.
     - Check if 373 is prime: 373 is prime.
     - Check if 3737 is prime: 3737 is not prime (divisible by 7).
     - Therefore, valid numbers are 3, 37, and 373.

   - **Starting with 5:**
     - The next digit must be 3 (or no next digit).
     - Valid numbers: 5, 53.
     - Check if 537 is prime: 537 is not prime (divisible by 3).
     - Therefore, valid numbers are 5 and 53.

   - **Starting with 7:**
     - The next digit must be 3 (or no next digit).
     - Valid numbers: 7, 73.
     - Check if 737 is prime: 737 is not prime (divisible by 11).
     - Therefore, valid numbers are 7 and 73.

3. **Count the valid numbers:**
   - From the above analysis, the valid numbers are:
     - Starting with 2: 2, 23.
     - Starting with 3: 3, 37, 373.
     - Starting with 5: 5, 53.
     - Starting with 7: 7, 73.
   - Total valid numbers: 2 + 3 + 2 + 2 = 9.

The final answer is $\boxed{9}$.