Since the formula is being applied twice (once with 3 and once with +3) to each prime generated, each prime may have at most two "children". So if p=2, then its children are 10*23 = 17 and 10*2+3 = 23. When p=167 there is but one child 10*1673=1667 because 10*167+3=1673 which is (7 * 239) and therefore not a prime. Essentially a tree of primes is being built which is at best binary:
.................3=1667 *10
.........................+3=16673
.........3=167 *10
..3=17 *10
........+3=173 *10
..................................3=173267
........................3=17327 *10
...........................................3=1732727
.................................+3=173273 *10
................+3=1733 *10
.........................+3=17333
2 *10
................3=2267
........3=227 *10
........................................................3=22726667
............................................3=2272667 *10
.................................3=227267 *10
.......................3=22727 *10
...............+3=2273 *10
..+3=23 *10
.........+3=233 *10
..............................................3=2332667 *10
.........................................................+3=23326673
...................................3=233267 *10
.........................3=23327 *10
.................+3=2333 *10
...................................3=233327
..........................+3=23333 *10
The 10p+3 tree for the root prime 2 is 8 generations deep and has a population of 28 nodes (including 2 itself).
The choice of 2 as the root of this tree, 10 as the coefficient and 3 as the +offset are not arbitrary. Performing this analysis for the first 1,000 primes for all combinations of coefficient (2 to 32) and offset (1 to 31) demonstrates that only 6p+5 (see A086321) and 10p+3 ever produce a tree with this many nodes on it. All other prime trees are smaller. 4p+3 produces a 25node tree when p=2, 12p+5 produces a 22node tree when p=2, and 28p+15 and 30p+7 produce 21node trees when p=953 and 13, respectively.
Note that the most populous tree formed need not be the deepest, since a single generation can produce 1 or 2 children for each parent. The deepest tree is 4p+3, which is 11 generations deep when p=2.
These results can be viewed at the link provided.
