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Since the formula is being applied twice (once with -5 and once with +5) to each prime generated, each prime may have at most two "children". So if p=2, then its children are 6*2-5 = 7 and 6*2+5 = 17. When p=37 there is but one child 6*37+5=227 because 6*37-5=217 which is (7 * 31) and therefore not a prime. Essentially a tree of primes is being built which is at best binary:
..........|-5=37 *6
..........|......|+5=227 *6
..........|..............|+5=1367
...|-5= 7 *6
...|......|..............|-5=1657
...|......|......|-5=277 *6
...|......|......|.......|+5=1667 *6
...|......|......|................|.........|-5=60037
...|......|......|................|+5=10007 *6
...|......|+5=47 *6
2 *6
...|.....................|-5=3457 *6
...|.....................|........|.........|-5=124477
...|.....................|........|+5=20747 *6
...|.............|-5=577 *6
...|.............|.......|+5=3467 *6
...|.............|................|+5=20807 *6
...|.............|..........................|+5=124847
...|......|-5=97 *6
...|......|......|.......|-5=3517 *6
...|......|......|.......|........|+5=21107
...|......|......|+5=587 *6
...|......|..............|........|-5=21157
...|......|..............|-5=3527 *6
...|+5=17 *6
..........|+5=107 *6
..................|.......|-5=3877
..................|+5=647 *6
The 6p+-5 tree for the root prime 2 is 7 generations deep and has a population of 28 nodes (including 2 itself).
The choice of 2 as the root of this tree, 6 as the coefficient and 5 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 and 10p+-3 (see A086322) ever produce a tree with this many nodes on it. All other prime trees are smaller. 4p+-3 produces a 25-node tree when p=2, 12p+-5 produces a 22-node tree when p=2, and 28p+-15 and 30p+-7 produce 21-node 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.
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