Problems & Puzzles: Puzzles

Puzzle 562.- Heleen's sequence

Jeff Heleen sent the following puzzle:

Start with any prime P. Create P-1 and P+1. Discard
any common factors from each then multiply the
remaining factors and add 1. If the new number is
prime iterate the process until the result is composite.

Ex. Start with P=11
Step 1: P-1=10, P+1=12
Step 2: 10 = 2 * 5; 12 = 2 * 2 * 3
Step 3: (2 * 3 * 5) + 1 = 31
Step 4: 31 is prime so iterate.

The entire process starting with 11 is
11 -> 31 -> 241 -> 14521 which is composite so
11 generates 2 primes, 31 and 241.

Starting with the prime 5 generates 5 primes:
5 -> 7 -> 13 -> 43 -> 463 -> 53593 -> 718052413 C

Complete the table. The starting prime may not
belong to a longer sequence. Ie, 7 generates 4
primes but 7 is already part of the 5 sequence.

 # of primes generated Start prime 2 11 3 4 5 5 6 7 8 9 10

Contributions came from Torbjörn Alm.

***

T. Alm wrote:

Generated 3
1  +991
2  +245521
3  +15070140361
4  +56777282625060302581

Generated 4
1  +317563
2  +25211564743
3  +158905749197620164013
4  +6312759282014240349757737674631255066043
5  +9962732438164236831056093175315081347598375985676130626325190341592238072919463

Gen        Start prime
2            11
3            991
4            317563
5            5

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