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A086322 Primes produced by repeated application of the formula p -> (10p +- 3) starting at the prime 2. 1
2, 17, 23, 167, 173, 227, 233, 1667, 1733, 2267, 2273, 2333, 16673, 17327, 17333, 22727, 23327, 23333, 173267, 173273, 227267, 233267, 233327, 1732727, 2272667, 2332667, 22726667, 23326673 (list; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=1..28.

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

Cf. A086319, A086321.

Sequence in context: A154963 A049562 A107137 * A164275 A284646 A270344

Adjacent sequences:  A086319 A086320 A086321 * A086323 A086324 A086325

KEYWORD

fini,full,nonn

AUTHOR

Chuck Seggelin (barkeep(AT)plastereddragon.com), Jul 24 2003

STATUS

approved

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Last modified January 20 10:58 EST 2020. Contains 331081 sequences. (Running on oeis4.)