The 17x+1 Map

Having recently written yet again about the Collatz trajectory, I was pleasantly surprised today to come upon a more generalised version of it. It goes by the name of the Px+1 map of which the Collatz trajectory is a specific example in which P=3. The Px+1 trajectory or map is an algorithm that states if x is divisible by any prime < P then divide out these primes one at a time starting with the smallest; otherwise multiply x by P and add 1.

My number for today is 25186 and it appears as an entry in OEIS A057534 that states:

• a(n+1) = a(n)/2 if 2|a(n)
• a(n)/3 if 3|a(n)
• a(n)/5 if 5|a(n)
• a(n)/7 if 7|a(n)
• a(n)/11 if 11|a(n)
• a(n)/13 if 13|a(n)
• else 17*a(n)+1

This is a particular example of the Px+1 map in which P=17 and this generates a sequence, part of which is shown below:

61, 1038, 519, 173, 2942, 1471, 25008, 12504, 6252, 3126, 1563, 521, 8858, 4429, 75294, 37647, 12549, 4183, 71112, 35556, 17778, 8889, 2963, 50372, 25186, 12593, 1799, 257, 4370, 2185, 437, 7430, 3715, 743, 12632, 6316, 3158, 1579, 26844, 13422, ...

I pondered long and hard over why 61 was chosen as the starting point for this sequence. Once a(0)=63 then the other terms follow according to the algorithm but why start there? I used Google Sheets to investigate the matter. As shown below, the sequence repeats after 84 steps. I'll leave the numbers vertical as it's easier to scan them.

61 prime
1038
519
173 prime
2942
1471 prime
25008
12504
6252
3126
1563
521 prime
8858
4429 (43 x 103)
75294
37647
12549
4183 (47 x 89)
71112
35556
17778
8889
2963 prime
50372
25186
12593
1799
257 prime
4370
2185
437 (19 x 23)
7430
3715
743 prime
12632
6316
3158
1579 prime
26844
13422
6711
2237 prime
38030
19015
3803 prime
64652
32326
16163
2309 prime
39254
19627 (19 x 1033)
333660
166830
83415
27805
5561 (67 x 83)
94538
47269 prime
803574
401787
133929
44643
14881 (23 x 647)
252978
126489
42163
3833 prime
65162
32581 (31 x 1051)
553878
276939
92313
30771
10257
3419
263 prime
4472
2236
1118
559
43 prime
732
366
183
61 prime

This still didn't explain why 61 was chosen as the starting point so I decided to investigate all the primes from 17 to 59 and from 67 up to 97. Here is a summary of my results:

17 terminates after 13 steps
19 terminates after 7 steps
23 terminates after 6 steps
29 terminates after 10 steps
31 terminates after 7 steps
37 terminates after 6 steps
41 terminates after 53 steps
43 repeats after 84 steps (joins 61 trajectory after 4 steps)
47 terminates after 8 steps
53 terminates after 56 steps (joins the 41 trajectory after 3 steps)
59 terminates after 13 steps
61 repeats after 84 steps
67 terminates after 12 steps
71 terminates after 15 steps
73 terminates after 11 steps
79 terminates after 9 steps
83 terminates after 36 steps
89 terminates after 9 steps
97 terminates after 6 steps

I thought it odd that 43 and 61 both repeated after 84 steps and closer investigation revealed that 43 is actually on the trajectory of 61 because of 43, 732, 366, 183, 61. So I guess 61 was chosen because it's the first sequence that loops instead of terminating. 43 also loops but actually joins the trajectory of 61 after 4 steps.

It would interesting be to explore what happens to larger primes under the 17x+1 map and also to experiment with other values of P, all possible of course with a spreadsheet.

My conjecture is that there is just the single 61 loop as shown above and that any number not in that loop must either enter it or else get reduced to 1 sooner or later.