[seqfan] Re: Why does this sequence make a staircase pattern?

[Hans Havermann]

EB: "I had thought of your explanation before asking the question and it does partly explain the staircase. But if it was the only reason, the least number in the 'step' would have the largest order, as every number generates an increasing sequence. But that isn’t always the case."

I don't want to think too hard about what you are saying (comprehension was never my strong suit) so, instead, I'll do a little hand-waving. The first plateau is {27,26} with the corresponding sequences of {27,35,39,41} and {26,27,35,39,41}, respectively. I guess this is what I had in mind when I proffered my explanation. The next plateau is {182,183} with corresponding sequences of {182,195,209,219,221,233} and {183,185,189,215,219,221,233}, respectively. Only the final three terms in each are the same.

Let me define a plateau as a series (>1) of numbers whose corresponding sequences end up at the same prime. So:

1 {27,26} -> 41
2 {182,183} -> 233

Continuing...

3 {319,280} -> 389
4 {1989,1985,1983,1922} -> 2417
5 {11760,13371} -> 15959
6 {15606,16659,15827,15732,15833,15210,15416,15707,15334,15251,15006,14812,14674,14787,14786} -> 18899
7 {283221,283091,301659,301655,294932,256000,303513} -> 336689
8 {292032,290950,297269,296865,295836,298348,301522,294872,300501,300639,300620,300627,300611,298779,293260,299965,300434} -> 334049

Note that, temporally, not all terms in #7 and #8 are consecutive. The first term of #8 occurs between the fourth and fifth terms of #7. This is what can happen when your final primes are too close. Anyways, continuing...

9 {4876656,4723056,4732922,4832843,4832643,4837869,4863407,4837395,4800090,4472160,4838466,4813876,4869295} -> 5149619
10 {9595188,9600350,9675103,9670001,9669999,9662264,9668803,9660470} -> 10609631
11 {13530943,13530832,13474183,13532419,13411321,13427620,13447444} -> 14405561
12 {18840921,18804988,18824932,18844033,18844519,18844510} -> 19365107
13 {46059481,46058293} -> 48545207
14 {248390199,248390150,248078957,248593619,248593615,248593612,247797164,248216210,248308029,248429344,248446096,247488441,247721100,248527331,248479672,248191952} -> 253692281
15 {516911157,518239445,518239443} -> 529616999
16 {615341209,615375467} -> 627470579
17 {717293892,716648869,716295841,716594005,716593987,716592382} -> 724418369
18 {779926646,780002264,775947645,780113799,779493578,779929006,780317203,780083215,780083182,780388991,780388416} -> 788554337

I count 125 "plateau" numbers thus far. Out of 137 points. That leaves 12 singletons (plateaus of length one), record-length sequences whose final term is (1 or) a prime. I may as well show these {#,final-sequence-term}:

{1,1}, {4,5}, {12,17}, {842,1013}, {1045,1367}, {1718,1907}, {5673,6659}, {8546,9629}, {55911,65687}, {137068,152459}, {98143757,100301237}, {588068633,600592829}.