Inspired by the question here, we did a few more simulations of numbers of some specific forms and noticed a pattern.

We consider the original $3n+1$ transform where we divide by $2$ if it's even and multiply $3$ and add $1$ if it's odd. The stopping times of the sequences are defined as the number of iterations it took to reach $1$ (which is believed to be finite for all natural numbers in this case).

We look at numbers of the form $2^n+1$ (as in the link above) and note that the stopping times create the following sequence:

$7, 5, 19, 12, 26, 27, 121, 122, 35, 36, 156, 113, 52, 53, 98, 99, 100, 101, 102, 72, 166, 167, 168, 169, 170, 171, 247, 173, 187, 188, 251, 252, 178, 179, 317, 243, 195, 196, 153, 154, 155, 156, 400, 326, 495, 496, 161, 162, 331, 332, 408, 471, 410, 411, 337, 338, 339, 340, 553, 479, 480, 481, 482, 483, 559, 560, 561, 562, 563, 564, 565, 566, 567, 568, 569, 570, 571, 572, 573, 574, 575, 576, 626, 578, 628, 629, 630, 631, 583, 584, 634, 635, 636, 637, 894, 895, 640, 641, 898, 643, 644, 645, 902, 903, 648, 649, 769, 907, 652, 653, 654, 655, 656, 657, 658, 915, 916, 917, 918, 919, 920, 921, 914, 923, 916, 917, 918, 919, 920, 921, 930, 923, 994, 995, 996, 997, 998, 999, 938, 1001, 1002, 1003, 1004, 1005, 1006, 1007, 1145, 1009, 1147, 1148, 1012, 1013, 1151, 1152, 1016, 1017, 1018, 1019, 1020, 1021, 1022, 1023, 1528, 1529, 1530, 1531, 1532, 1533, 1534, 1535, 1169, 1537, 1445, 1446, 1447, 1448, 1542, 1543, 1544, 1545, 1546, 1547, 1548, 1549, 1550, 1551, 1552, 1553, 1554, 1555, 1556, 1557, 1527, 1528, 1560, 1561, 1562, 1563, 1533,\ \ldots$

The curious fact is that there seems to exist arithmetical sequences with common difference $1$ and of various lengths existing in the sequence. In this sequence shown, the largest such sequence starts at $559$ and ends at $576$ with a length of $18$.

Next we looked at numbers of the form $3^n+1$ and noted the following sequence of stopping times:

$2, 6, 18, 110, 21, 95, 32, 75, 74, 42, 134, 133, 132, 131, 143, 204, 128, 189, 139, 94, 93, 260, 427, 90, 257, 393, 330, 254, 253, 389, 388, 387, 461, 460, 459, 458, 457, 456, 455, 454, 453, 452, 500, 499, 449, 497, 496, 751, 494, 493, 492, 747, 490, 745, 488, 487, 486, 741, 740, 739, 738, 737, 728, 727, 726, 725, 794, 793, 792, 791, 790, 789, 788, 787, 923, 785, 921, 783, 782, 781, 780, 1283, 1282, 1281, 1280, 1279, 1185, 1184, 1276, 1275, 1274, 1273, 1272, 1271, 1270, 1269, 1237, 1267, 1266, 1234, 1264, 1263, 1231, 1230, 1229, 1228, 1227, 1226, 1225, 1255, 1422, 1222, 1420, 1419, 1418, 1417, 1248, 1216, 1246, 1214, 1213, 1212, 1211, 1409, 1408, 1407, 1406, 1343, 1342, 1403, 1733, 1339, 1338, 1337, 1336, 1335, 1727, 1333, 1681, 1724, 1723, 1678, 1677, 1720, 1675, 1718, 1717, 1716, 1715, 1714, 1713, 1712, 1711, 1710, 1709, 1708, 1707, 1706, 1705, 1704, 1703, 1702, 1701, 1700, 1699, 1698, 1697, 1696, 1695, 1694, 1693, 1692, 1691, 1690, 1707, 1688, 1687, 1686, 1703, 1702, 1701, 1700, 1681, 1680, 1679, 1678, 1695, 1676, 1675, 1692, 1691, 1690, 1689, 2585, 1687, 1686, 2334, 2333, 2332,\ \ldots$

Here also, we notice that there exist arithmetic sequences with common difference $-1$ and again of various lengths in the sequence. The largest such sequence here starts at $1718$ till $1690$ with a length of $29$.

We did these simulations a few more times with more and more numbers and sequences and essentially noted the following:

So we tried combining the two observations and guessed that numbers of the form $6^n+1$ should create sequences would have arithmetic sequences with common difference $0$. And yes it does so. The below sequence shows it:

$16, 21, 26, 101, 83, 83, 145, 145, 220, 158, 145, 207, 114, 114, 450, 114, 357, 357, 282, 419, 419, 494, 494, 494, 494, 494, 494, 494, 543, 494, 543, 799, 799, 543, 543, 799, 543, 543, 799, 799, 791, 791, 791, 791, 861, 861, 861, 861, 998, 998, 998, 861, 861, 861, 1365, 1365, 998, 1272, 1365, 1365, 1365, 1365, 1365, 1365, 1365, 1365, 1365, 1365, 1334, 1334, 1334, 1334, 1365, 1533, 1533, 1533, 1533, 1365, 1365, 1334, 1533, 1334, 1533, 1533, 1471, 1471, 1864, 1471, 1471, 1471, 1864, 1471, 1864, 1864, 1820, 1864, 1864, 1864, 1864, 1864, 1864, 1864, 1864, 1864, 1864, 1864, 1864, 1864, 1864, 1864, 1864, 1864, 1864, 1864, 1864, 1864, 1864, 1864, 1882, 1882, 1882, 1864, 1864, 1864, 1864, 1864, 1882, 1882, 2779, 1882, 2531, 2531, 2779, 1882, 1882, 2779, 2779, 2779, 2779, 2779, 2779, 2779, 2779, 2779, 2779, 2805, 2805, 2805, 2779, 2779, 2779, 2779, 2779, 2779, 2779, 2779, 2779, 2779, 2779, 2779, 2779, 2779, 2779, 2779, 2779, 2779, 2779, 2779, 2779, 2779, 2779, 2779, 2779, 2779, 2779, 2779, 2779, 3203, 2779, 2779, 3203, 2779, 2779, 3203, 3203, 3203, 3932, 3932, 3932, 3932, 3932, 3203, 3932, 3932, 3203, 3203, 3203, 3203, 4586,\ \ldots$

Also:

The final thing that we observed (sorry for not having a table ready, I will edit it as soon as possible) is that any number of the form $(2^a3^b)+1$ had stopping time sequences with the existence of arithmetic progressions with common difference $a-b$.

Claim: Any number of the form $(2^a3^b)+1$ has stopping time sequences with the existence of arithmetic progressions with common difference $a-b$.

We know this is true, but a proof eludes us.

Following everything we found, we made a conjecture :

Strong Conjecture: If the Collatz conjecture is true then the sequence of stopping times of the Collatz sequence for numbers of the form $(2^a3^b)^n +1$ has arbitrarily long arithmetic sequences and with common difference $a-b$.

We were not successful in proving or disproving the result, neither did we find any literature on this.

Any input on the topic would be highly appreciated.

Edit: I added a claim separately to highlight a part I just mentioned in passing.

We believe that there may not be arbitrarily long progressions because clearly we were missing a lot of values in the length sequences. But they did seem to be unbounded (no upper limit on the length of the progressions). So we weakened the conjecture a bit:

Weak Conjecture: If the Collatz conjecture is true then the sequence of stopping times of the Collatz sequence for numbers of the form $(2^a3^b)^n +1$ has arithmetic sequences with no length bound and with common difference $a-b$.

Evidence for the weakening:

For numbers of the form $6^n+1$, the sequence of length of progressions (what progression lengths are there in the stopping time sequences) looked like this:

For the first $1000$ terms:

$1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 27, 29, 38, 48, 51, 52$

For the first $2000$ terms:

$1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 27, 28, 29, 36, 38, 45, 46, 48, 51, 52, 59, 110$

For the first $10000$ terms:

$1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 31, 36, 38, 41, 45, 46, 48, 50, 51, 52, 54, 55, 56, 58, 59, 60, 61, 65, 71, 72, 73, 74, 78, 82, 83, 86, 92, 99, 101, 102, 107, 109, 110, 111, 116, 126, 130, 132, 133, 160, 161, 163, 172, 180, 182, 184, 185, 186, 189, 220, 248, 271, 337$

We are clearly missing $12$ and $24$. Probably $30$ is also missing. We are not sure if these are actually missing, but appears to be so. If indeed they are missing, then the strong version is false. If not i.e. if the values will be available as the number of terms increase, then it's true.

But as of this moment, both the conjectures are open (possibly).

So the main questions to move forward will be to the answer the following:

- Why are their arithmetic sequences in the stopping times of these specific numbers?
- Can we prove the claim?
- Are the conjectures true?

(I will be editing the question with more data and possibly some graphs as soon as I have some more time on hand. Till then, please let us know of your thoughts. Thank you).

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