|a-b| divides the
concatenation [a,b]
Hello
SeqFans,
S =
1,144,146,153,156,160,165,176,184,197,274,288,294,315,324,336,352,374, ...
We
want:
1) S
to be strictly increasing
2) all first diff to be different one from another and not yet
present in S
3) a(n+1) to be the smallest integer such that |a(n)-a(n+1)|
divides the concatenation [a(n),a(n+1)]
Here
is how we get S, starting with 1:
S = 1,
144,146,153,156,160,165,176,184,197,274,288,294,315,324,336,352,374, ...
1st
dif: 143 2
7 3 4 5
11 8 13
77 14 6
21 9 12
16 22
143 is the smallest integer not yet present and dividing 1144 (=8)
2 is the smallest integer not
yet present and dividing 144146 (=72073)
7 is the smallest integer not
yet present and dividing 146153 (=20879)
3 is the smallest integer not
yet present and dividing 153156 (=51052)
4 is the smallest integer not
yet present and dividing 156160 (=39040)
5 is the smallest integer not
yet present and dividing 160165 (=32033)
11 is the smallest integer not
yet present and dividing 165176 (=15016)
...
---
If
we drop the "strictly increasing" constraint, we’ll get T (which is
an incredible nightmare to calculate by hand -- WARNING, the correct T seq
is not this one -- see Doug’s
comment below):
T = 1,
144, 43, 134, 108, 9, 6, 4,
158, ...
1st
dif: 143 101 91
26 99 3
2 154 ...
More
terms for S & T (if of interest)?
Best,
É.
__________
[Douglas McNeil]:
Assuming I
understand correctly:
sage: S
[1, 144, 146, 153,
156, 160, 165, 176, 184, 197, 274, 288, 294, 315, 324, 336, 352, 374, 391, 414,
432, 456, 475, 500, 510, 525, 558, 584, 612, 646, 684, 720, 740, 775, 806, 868,
912, 951, 1024, 1056, 1104, 1150, 1200, 1230, 1271, 1408, 1472, 1564, 1632,
1683, 1782, 1809, 1876, 2010, 2211, 2430, 2475, 2530, 2640, 2680, 2948, 3240,
3294, 3355, 3660, 3720, 3813, 3936, 4018, 4067, 4150, 4316, 4368, 4410, 4473,
4544, 4686, 4752, 4824, 5226, 5304, 5406, 5459, 5562, 5768, 5824, 5915, 5980,
6095, 6360, 6466, 6588, 6696, 6820, 6975, 7050, 7097, 7248, 7550, 7915, 8600]
which I
think agrees with your values, and
sage: T
[1, 144, 43, 120,
80, 60, 5, 390, 87, 58, 56, 42, 9, 6, 160, 108, 72, 48, 4, 186, 124, 93, 100,
75, 90, 103, 114, 76, 57, 148, 111, 132, 104, 78, 117, 180, 110, 88, 96, 64,
176, 192, 126, 175, 140, 130, 195, 300, 98, 384, 153, 102, 68, 85, 204, 216,
162, 135, 234, 252, 168, 315, 150, 200, 420, 224, 432, 159, 106, 477, 906, 360,
330, 209, 342, 513, 266, 684, 152, 460, 138, 92, 69, 230, 345, 598, 260, 624,
288, 444, 185, 592, 518, 999, 222, 1044, 225, 270, 459, 238, 425]
Note I find T(4)=120.
Doug
--
Department of
Earth Sciences
University of Hong
Kong
__________
Beautiful!
Many thanks!
Best,
É.