1. 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.

2.

3. Just out of curiosity, but you do know that the easy way to compute the recursive sum of digits is just to take the number mod 9?

4. Didn't know. So it is remainder after dividing by 9?

5. Yeah, with the caveat that if you would get a 0, the answer is really 9 (unless the original number was 0).

6. The problem is to prove item 3 for all primes. I can't see a starting point.

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