OFFSET
1,1
COMMENTS
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*2-3 = 17 and 10*2+3 = 23. When p=167 there is but one child 10*167-3=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 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.
LINKS
C. Seggelin, Deepest Prime Trees
FORMULA
a(n) = (10 * a(n-m)) - 3 or (10 * a(n-m)) + 3.
MATHEMATICA
a[1] = {2}; a[n_] := Union[ Join[ a[n - 1], Select[ Flatten[{10*a[n - 1] - 3, 10*a[n - 1] + 3}], PrimeQ[ # ] &]]]; a[8]
CROSSREFS
KEYWORD
fini,full,nonn
AUTHOR
Chuck Seggelin (barkeep(AT)plastereddragon.com), Jul 24 2003
STATUS
approved