[seqfan] Riecaman

[hv at crypt.org]

Here is a Recaman-inspired sequence with apparent ties to Riemann.

The main purpose of this is to ask why on earth we end up with a series
of primes whose successive ratios converge to 9.

Define a mapping n_{i-1} -> n_i as:
  n_{i-1} + p_i    if p_i > n_{i-1}
  n_{i-1} - p_i    otherwise
with a given starting point n_0, and where p_i is the i'th prime.

Define a(n) as the least positive k such that n_k is 0 when we set n_0 = n,
or as 0 if no such k exists.

I've calculated most values of a(0) .. a(100), (see below); the missing ones
are for n in { 6 16 20 30 42 50 51 56 70 71 76 84 85 90 92 }, and there
things get interesting. If k exists for any of these, it is at least 1.5e10.

Several of those hard ones collapse to identical trajectories early on:
  6, 16, 30, 56, 90
  20, 42, 76
  50, 84
  51, 85
.. so a(6) = a(16) etc. Taking the first of such sets as "primitive",
that leaves primitives { 6 20 50 51 70 71 92 }.

The shape of the trajectories is that we alternately add and subtract,
with the net effect that n_{i+2} is smaller than n_i by the prime difference,
until we reach a local mininum that is either zero (terminating the
process) or too small for the prime difference causing us to add twice
in a row. When we hit a local minimum we also switch the parity of
the lower of each pair of terms, and obviously we can't hit zero when
that's odd.

Checking the even local minima for the trajectory of 6, for example,
gives:
n  n_i  i          p_i
6    2  5          13
6    8  29         113
6    4  199        1223
6    2  1355       11197
6    8  9589       99971
6   30  70579      890377
6   60  539961     8002847
6    4  4228745    72001673
6   38  33690443   647909833
6   16  272003821  5830319399
6   16  2219823175 52470123707

I looked at this mostly to try and understand whether I should expect
0 values of the sequence to exist - my conjecture is no - but looking
at the ratio of the p_i for those local minima, they turn out to be
converging on something astonishingly close to 9. For n_0 = 6 we get:

8.69230769230769 (= 113/13)
10.8230088495575
9.15535568274734
8.92837367151916
8.90635284232427
8.98815557904124
8.99700731502177
8.99853858951305
8.99865861273323
8.99952817610636

The ratios in other examples look similar, eg for 71:

8.01910828025478
8.90627482128674
9.09542495317934
9.09126653397002
9.01309546792409
9.00090632116533
9.00247399195550
9.00106389627424
9.00051634900210

That makes no sense to me. I hope someone else can explain it.

Hugo van der Sanden
---
0 3
1 2
2 1
3 6
4 3
5 2
6 unknown
7 4
8 69
9 6
10 3
11 58
12 23
13 10
14 5
15 12
16 unknown
17 4
18 69
19 6
20 unknown
21 8
22 21
23 56
24 369019
25 58
26 23
27 10
28 5
29 12
30 unknown
31 14
32 7
33 16
34 37
35 18
36 9
37 122
38 11
39 30
40 69
41 6
42 unknown
43 8
44 21
45 56
46 369019
47 58
48 23
49 10
50 unknown
51 unknown
52 25
53 70
54 27
55 12
56 unknown
57 14
58 7
59 16
60 37
61 18
62 9
63 122
64 11
65 30
66 69
67 8458
68 13
69 36
70 unknown
71 unknown
72 15
73 44
74 107
75 46
76 unknown
77 8
78 21
79 56
80 369019
81 58
82 23
83 10
84 unknown
85 unknown
86 25
87 70
88 27
89 12
90 unknown
91 14
92 unknown
93 4234
94 33
95 4336
96 233
97 16
98 37
99 18
100 9