I have found something about primes directly from the numbers.

Except for 2 and 3 there are just six types of other primes. The idea is to let the primes order themselves into subsets as follows.

1) Compute the recursive sum of digits, then the 6 types are those with matching recursive sum of digits. See the following list:

primes: 5 7 11 13 17 19

SigmaR: 5 7 2 4 8 1

Kind: 5 7 11 13 17 19

2) List the primes in their order together with their kind. Then patterns can be computed from every series of increasing primes (according to the kind). The list starts:

Primes: 23, 29, 31, 37 | 41, 43, 47, 53 |

Kind:___5 11 13 19__| 5 7 11 17____|

Pattern: []*[][]*[]____| [][][]*[]*_____|

where the pattern essentially compares the kind with 5, 7, 11, 13, 17, 19 with symbol [] if the kind occurs in the string and * if not. Matching patterns defines subsets satisfying item 3.

3) Patterns that match (not singletons like *[]****) have difference of their first members being divisable by 6.

The first matching pattern with [][][]*[]* is:

Primes: 347, 349, 353, 359

Kind: 5 7 11 17

and 347 - 41 = 306 and is divisable by 6. Item 3 was tested untill 1733.