The fun in numbers

A friend of mine gave me this riddle.

Find the next number of the sequence:

293, 313, 331, ...

Let's see who can do better than I did.

Hint:

There is a new number:

263, 293, 313, 331, ...

• Did your friend give any additional numbers or clues? With three numbers there will be many sequences that may fit, but not be the intended answer. for example: rot13(svefg nqq gjragl, gura nqq rvtugrra, nqq fvkgrra naq fb ba) Commented May 25, 2020 at 6:57
• Yes, but he gave me the previous number' not the next one. I will edit the question and add the new number. Commented May 25, 2020 at 7:01
• does this is related to 10-happy-prime ? Commented May 25, 2020 at 14:28

The sequence is:

A list of happy prime numbers

So a longer list includes:

7, 13, 19, 23, 31, 79, 97, 103, 109, 139, 167, 193, 239, 263, 293, 313, 331, 367, 379, 383, 397, 409, 487.

• @omri klein Make sure you not to tell your friend that you had help from a community with solving it Commented May 25, 2020 at 18:15

The sequence of prime numbers follows the pattern below

›! The next number is 337.

263 4x+3

293 4x+1

313 4x+1

331 4x+3

337 4x+1

• Nice idea, but check out the last answer, it's the right answer. Commented May 25, 2020 at 19:11
• From the number you gave I drew the conclusion one prime 4x+3 two primes 4x+1 one prime 4x+3 and two primes 4x+1. Commented May 25, 2020 at 20:39