Problems & Puzzles: Puzzles

Puzzle 562.- Heleen's sequence

Jeff Heleen sent the following puzzle:

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.

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.

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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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