Slide my digits and divide
Three
sequences with sliding divider-digits
A. Lexicographically earliest sequence of distinct positive terms such that any
digit “d” sliding “d” terms to the right divides the last term of the slip – except
for the digit 0 that does not move and does nothing.
B. Lexicographically
earliest sequence of distinct nonnegative terms such that any digit “d” sliding
“d” terms to the left divides the last term of the slip – except for the digit
0 that does not move and does nothing
C. Lexicographically
earliest sequence of distinct nonnegative terms such that any digit “d” sliding
“d” terms either to the right or to the left divides the last term of the slip –
except for the digit 0 that does not move and does nothing.
A = 1,
2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 24, 16, 36, 17, 20, 18, 22, 30,
42, 19, 84, 21, 28, 32, 40, …
Check
The
1st digit of A is 1, slides 1 term to the right and divides the term
2:
A = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11,
12, 13, 14, 15, 24, 16, 36, 17, 20, 18, 22, 30, 42, 19, 84, 21, 28, 32, 40, …
The next
digit is 2, slides 2 terms to the right and divides the term 4:
A = 1,
2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12,
13, 14, 15, 24, 16, 36, 17, 20, 18, 22, 30, 42, 19, 84, 21, 28, 32, 40, …
The next
digit is 3, slides 3 terms to the right and divides the term 6:
A = 1,
2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14,
15, 24, 16, 36, 17, 20, 18, 22, 30, 42, 19, 84, 21, 28, 32, 40, …
The next
digit is 4, slides 4 terms to the right and divides the term 8:
A = 1,
2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15,
24, 16, 36, 17, 20, 18, 22, 30, 42, 19, 84, 21, 28, 32, 40, …
The next
digit is 5, slides 5 terms to the right and divides the term 10:
A = 1,
2, 3, 4, 5, 6, 7, 8,
9, 10, 11, 12, 13, 14, 15,
24, 16, 36, 17, 20, 18, 22, 30, 42, 19, 84, 21, 28, 32, 40, …
…
The
hereunder yellow digit 8 slides 8 terms to the right and divides the term 24:
A = 1,
2, 3, 4, 5, 6, 7, 8,
9, 10, 11, 12, 13, 14, 15, 24,
16, 36, 17, 20, 18, 22, 30, 42, 19, 84, 21, 28, 32, 40, …
The next
digit is 9, slides 9 terms to the right and divides the term 36:
A = 1,
2, 3, 4, 5, 6, 7, 8, 9,
10, 11, 12, 13, 14, 15, 24, 16, 36, 17, 20, 18, 22, 30, 42, 19, 84, 21, 28, 32, 40, …
The
next digit is 1, slides 1 term to the right and divides the term 11:
A = 1,
2, 3, 4, 5, 6, 7, 8, 9, 10,
11, 12, 13, 14, 15, 24,
16, 36, 17, 20, 18, 22, 30, 42, 19, 84, 21, 28, 32, 40, …
The next
digit is 0, which does not move and does nothing:
A = 1,
2, 3, 4, 5, 6, 7, 8, 9, 10,
11, 12, 13, 14, 15, 24, 16, 36, 17, 20, 18, 22, 30, 42, 19, 84, 21, 28, 32, 40,
…
The
next digit is 1, slides 1 term to the right and divides the term 12:
A = 1,
2, 3, 4, 5, 6, 7, 8, 9, 10, 11,
12, 13, 14, 15, 24, 16,
36, 17, 20, 18, 22, 30, 42, 19, 84, 21, 28, 32, 40, …
The
next digit is 1, slides 1 term to the right and also divides the term 12:
A = 1,
2, 3, 4, 5, 6, 7, 8, 9, 10, 11,
12, 13, 14, 15, 24, 16,
36, 17, 20, 18, 22, 30, 42, 19, 84, 21, 28, 32, 40, …
The
next digit is 1, slides 1 term to the right and divides the term 13:
A = 1,
2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14,
15, 24, 16, 36, 17, 20, 18, 22, 30, 42, 19, 84, 21, 28, 32, 40, …
The
next digit is 2, slides 2 terms to the right and divides the term 14:
A = 1,
2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14,
15, 24, 16, 36, 17, 20, 18, 22, 30, 42, 19, 84, 21, 28, 32, 40, …
The
next digit is 1, slides 1 term to the right and also divides the term 14:
A = 1,
2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15,
24, 16, 36, 17, 20, 18, 22, 30, 42, 19, 84, 21, 28, 32, 40, …
The
next digit is 3, slides 3 terms to the right and divides the term 24:
A = 1,
2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 24,
16, 36, 17, 20, 18, 22, 30, 42, 19, 84, 21, 28, 32, 40, …
The
next digit is 1, slides 1 term to the right and divides the term 15:
A = 1,
2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 24, 16, 36, 17, 20, 18, 22, 30, 42, 19, 84, 21, 28, 32, 40, …
The
next digit is 4, slides 4 terms to the right and divides the term 36:
A = 1,
2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 24, 16, 36, 17, 20, 18, 22, 30, 42, 19, 84, 21, 28, 32, 40, …
Etc.
______________________________
B =
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 100, 17, 13, 14, 40, 15, 20, 101, 21,
50, 41, 22, 18, 102, 25, 23, 30, …
Check
The
1st digit of B is 0, which does not move and does nothing:
B = 0, 1, 2, 3, 4, 5, 6, 7,
8, 9, 10, 11, 12, 100, 17, 13, 14, 40, 15, 20, 101, 21, 50, 41, 22, 18, 102,
25, 23, 30, …
The next
digit is 1, slides 1 term to the left and divides the term 0:
B = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 100,
17, 13, 14, 40, 15, 20, 101, 21, 50, 41, 22, 18, 102, 25, 23, 30, …
The next
digit is 2, slides 2 terms to the left and also divides the term 0:
B = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 100,
17, 13, 14, 40, 15, 20, 101, 21, 50, 41, 22, 18, 102, 25, 23, 30, …
The next
digit is 3, slides 3 terms to the left and also divides the term 0:
B = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,
11, 12, 100, 17, 13, 14, 40, 15, 20, 101, 21, 50, 41, 22, 18, 102, 25, 23, 30, …
…
The
hereunder yellow digit 1 slides 1 term to the left and divides the term 9:
B =
0, 1, 2, 3, 4, 5, 6, 7, 8, 9,
10, 11, 12, 100,
17, 13, 14, 40, 15, 20, 101, 21, 50, 41, 22, 18, 102, 25, 23, 30, …
The
next digit is 0, which does not move and does nothing:
B =
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 100, 17, 13, 14, 40, 15, 20, 101, 21, 50, 41, 22, 18,
102, 25, 23, 30, …
The next
digit is 1, slides 1 term to the left and divides the term 10:
B =
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,
11, 12, 100, 17,
13, 14, 40, 15, 20, 101, 21, 50, 41, 22, 18, 102, 25, 23, 30, …
The next
digit is 1, slides 1 term to the left and also divides the term 10:
B =
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,
11, 12, 100, 17,
13, 14, 40, 15, 20, 101, 21, 50, 41, 22, 18, 102, 25, 23, 30, …
The next
digit is 1, slides 1 term to the left and divides the term 11:
B =
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12,
100, 17, 13, 14, 40, 15, 20, 101, 21, 50, 41, 22, 18, 102, 25, 23, 30, …
The next
digit is 2, slides 2 terms to the left and also divides the term 10:
B =
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,
11, 12, 100, 17,
13, 14, 40, 15, 20, 101, 21, 50, 41, 22, 18, 102, 25, 23, 30, …
The next
digit is 1, slides 1 term to the left and divides the term 12:
B =
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 100,
17, 13, 14, 40, 15, 20, 101, 21, 50, 41, 22, 18, 102, 25, 23, 30, …
The
next two digits are 0s and we leave them in peace:
B =
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 100, 17, 13, 14, 40, 15, 20, 101, 21, 50, 41, 22,
18, 102, 25, 23, 30, …
The next
digit is 1, slides 1 term to the left and divides the term 100:
B =
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 100, 17, 13, 14, 40, 15, 20, 101, 21, 50, 41, 22, 18, 102, 25, 23,
30, …
The next
digit is 7, slides 7 terms to the left and divides the term 7:
B =
0, 1, 2, 3, 4, 5, 6, 7,
8, 9, 10, 11, 12, 100, 17,
13, 14, 40, 15, 20, 101, 21, 50, 41, 22, 18, 102, 25, 23, 30, …
The next
digit is 1, slides 1 term to the left and divides the term 17:
B =
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 100, 17, 13, 14, 40, 15, 20, 101, 21, 50, 41, 22, 18, 102, 25, 23, 30, …
The next
digit is 3, slides 3 terms to the left and divides the term 12:
B =
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 100, 17, 13,
14, 40, 15, 20, 101, 21, 50, 41, 22, 18, 102, 25, 23, 30, …
The next
digit is 1, slides 1 term to the left and divides the term 13:
B =
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 100, 17, 13, 14, 40, 15, 20, 101, 21, 50, 41, 22, 18, 102, 25, 23, 30, …
The next
digit is 4, slides 4 terms to the left and also divides the term 12:
B =
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 100, 17, 13, 14, 40, 15, 20, 101, 21, 50, 41, 22, 18, 102, 25, 23, 30, …
The next
digit is 4, slides 4 terms to the left and divides the term 100:
B =
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 100, 17, 13, 14, 40, 15, 20, 101, 21, 50, 41, 22, 18, 102, 25, 23,
30, …
…
The
hereunder yellow digit 8 slides 8 terms to the left and divides the term 40:
B =
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 100, 17, 13, 14, 40, 15, 20, 101, 21, 50, 41,
22, 18, 102, 25,
23, 30, …
Etc.
(work
in progress)
(going
to bed now – hoping I did not leave too many mistakes above)
(will
compute by hand the seq C tomorrow)
____________________
Next morning update
Jean-Marc Falcoz was quick to extend A and B and send the according (beautiful) graphs::
A = 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,24,16,36,17,20,18,22,30,42,19,84,21,28,32,40,26,27,34,72,23,48,60,33,35,44,63,39,54,120,45,38,96,25,108,50,81,29,90,56,55,99,64,31,80,150,144,117,66,37,180,41,168,43,126,52,210,68,192,46,240,47,58,132,76,216,51,270,49,62,264,420,88,70,112,156,78,135,53,57,840,102,65,140,160,59,77,100,288,61,300,67,69,105,153,114,104,138,630,200,204,162,74,82,228,128,92,174,86,91,71,136,152,176,252,171,184,1050,189,330,196,73,315,75,312,87,336,225,175,198,98,79,1260,280,94,154,83,208,504,324,360,224,276,540,232,248,186,348,390,308,93,372,222,384,408,256,111,432,364,234,165,456,480,116,246,220,600,258,124,282,528,106,660,85,89,320,95,552,720,110,118,400,440,207,97,756,148,101,272,103,560,294,147,306,296,700,396,107,109,378,342,113,462,243,588,414,672,576,318,260,115,354,960,546,231,780,520,444,450,366,119,900,...
A – 10000 terms
B – 1000 terms
C = 0,1,2,3,4,5,6,7,8,9,10,11,12,100,70,13,24,14,90,15,20,700,16,21,25,40,30,99,210,300,18,22,33,28,50,36,126,41,120,35,72,336,42,17,60,19,68,630,26,27,420,110,102,66,144,23,56,130,96,162,111,200,48,32,108,114,84,234,29,80,168,88,112,31,104,44,192,288,128,184,34,101,46,444,180,1008,208,224,132,52,62,204,240,248,160,124,1020,64,106,140,222,152,82,312,150,118,270,43,122,55,216,188,116,1120,45,92,414,134,228,320,720,212,246,94,54,39,600,308,360,190,141,51,666,53,252,1000,810,91,1260,310,156,37,201,38,456,170,612,93,1204,330,264,1112,840,202,220,74,1044,105,61,380,304,121,525,1014,504,103,113,3000,408,660,63,47,148,69,65,816,384,71,133,390,1104,606,261,232,448,76,621,402,1080,142,123,372,1848,516,213,49,344,303,1610,1116,432,211,172,354,828,276,306,332,230,294,1002,426,2016,292,302,136,146,462,1026,174,280,900,1332,888,324,135,154,117,480,161,1620,800,244,57,196,450,528,221,340,808,1050,1110,164,153,328,115,540,768,1140,131,400,145,624,510,616,352,440,1010,1200,166,1440,73,58,81,468,1500,83,1030,1148,129,1560,1016,392,109,424,75,1224,10110,1188,77,500,333,441,601,138,95,2072,309,1012,1170,690,119,1024,301,522,151,12600,311,272,3330,1710,125,342,370,158,231,1680,10010,159,290,1001,250,576,710,512,203,410,7110,1062,2100,812,322,78,206,546,226,79,702,176,21000,1032,1122,214,1134,1302,6111,396,636,181,321,147,474,366,1206,912,2232,1232,266,1344,642,241,1212,348,1728,236,672,284,1146,242,260,3024,1128,1092,186,520,1230,262,602,282,960,502,2160,318,1242,67,1320,5000,1068,603,86,1944,315,191,618,804,2112,155,920,388,990,169,2520,544,182,205,218,5040,610,171,225,404,10000,163,1272,580,924,215,194,930,460,1022,1380,712,412,1152,1404,1422,1476,314,1400,268,1164,608,492,245,904,714,648,10200,560,832,472,1602,588,2250,1248,98,238,3360,620,177,696,401,488,1056,4104,820,2436,334,880,1314,552,1028,438,592,411,1410,680,143,860,11000,1512,256,2124,254,1488,430,640,708,1800,892,316,350,1824,165,882,13000,2040,992,2070,1434,416,107,1632,508,428,861,2142,149,4200,313,1088,30000,744,752,243,274,1548,1236,1160,1288,1428,1040,406,534,2400,940,732,728,4440,564,251,223,364,1540,10710,2280,484,1004,1350,127,452,1176,1100,1280,464,1210,1141,594,4004,281,2052,1808,1180,137,421,470,1494,1520,1220,928,5152,422,2240,442,1042,1980,1124,872,2088,704,524,233,1102,1144,12096,1880,2128,824,1114,1142,654,724,1108,792,1208,1240,1266,1202,1414,436,11200,801,1296,844,434,570,1011,1704,1308,1312,85,936,4011,446,1113,185,100000,1368,2000,780,1530,157,323,1416,518,2880,910,1424,1211,1290,550,476,2032,1072,590,2700,1021,2004,17010,1150,385,368,10101,1125,195,770,331,972,15000,5280,235,1034,490,405,1155,1060,1908,1328,255,530,1554,774,219,305,31000,9000,1101,207,455,454,59,1161,1888,1580,13230,1070,494,2010,505,2502,536,1712,511,1590,4000,2610, ...
Thank you, Jean-Marc – this is a marvel (three marvels!-)
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