Damn DAM madness

(Dall-e creation)
 
DAM (digits’ arithmetic mean) TABLE
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The DAM of 0 is arbitrarily given to numbers that do not have an integer DAM (see A180157): 10, 12, 14, 16, 18, 21, 23, 25, 27, 29, 30, 32, 34, 36, 38, 41, 43, 45, 47, 49, 50, 52, 54, 56, 58, 61, 63, 65, 67, 69, 70, 72, 74, 76, 78, 81, 83, 85, 87, 89, 90, 92, 94, 96, 98, 100, 101, 103, 104, 106, 107, 109, 110, 112, 113, 115, 116, 118, 119, 121, 122, 124, 125, …
Numbers having a DAM of 1 (see A061384): 1, 11, 20, 102, 111, 120, 201, 210, 300, 1003, 1012, 1021, 1030, 1102, 1111, 1120, 1201, 1210, 1300, 2002, 2011, 2020, 2101, 2110, 2200, 3001, 3010, 3100, 4000, 10004, 10013, 10022, 10031, 10040, 10103, 10112, 10121, 10130, 10202, 10211, …
Numbers having a DAM of 2 (see A061385): 2, 13, 22, 31, 40, 105, 114, 123, 132, 141, 150, 204, 213, 222, 231, 240, 303, 312, 321, 330, 402, 411, 420, 501, 510, 600, 1007, 1016, 1025, 1034, 1043, 1052, 1061, 1070, 1106, 1115, 1124, 1133, 1142, 1151, 1160, 1205, 1214, 1223, 1232, 1241, 1250, 1304, …
Numbers having a DAM of 3 (see A061386): 3, 15, 24, 33, 42, 51, 60, 108, 117, 126, 135, 144, 153, 162, 171, 180, 207, 216, 225, 234, 243, 252, 261, 270, 306, 315, 324, 333, 342, 351, 360, 405, 414, 423, 432, 441, 450, 504, 513, 522, 531, 540, 603, 612, 621, 630, 702, 711, 720, 801, 810, 900, 1029, ...
Numbers having a DAM of 4 (see A061387): 4, 17, 26, 35, 44, 53, 62, 71, 80, 129, 138, 147, 156, 165, 174, 183, 192, 219, 228, 237, 246, 255, 264, 273, 282, 291, 309, 318, 327, 336, 345, 354, 363, 372, 381, 390, 408, 417, 426, 435, 444, 453, 462, 471, 480, 507, 516, 525, 534, 543, 552, 561, 570, 606, ...
Numbers having a DAM of 5 (see A061388): 5, 19, 28, 37, 46, 55, 64, 73, 82, 91, 159, 168, 177, 186, 195, 249, 258, 267, 276, 285, 294, 339, 348, 357, 366, 375, 384, 393, 429, 438, 447, 456, 465, 474, 483, 492, 519, 528, 537, 546, 555, 564, 573, 582, 591, 609, 618, 627, 636, 645, 654, 663, 672, 681, ... 
Numbers having a DAM of 6 (see A061423): 6, 39, 48, 57, 66, 75, 84, 93, 189, 198, 279, 288, 297, 369, 378, 387, 396, 459, 468, 477, 486, 495, 549, 558, 567, 576, 585, 594, 639, 648, 657, 666, 675, 684, 693, 729, 738, 747, 756, 765, 774, 783, 792, 819, 828, 837, 846, 855, 864, 873, 882, 891, 909, …
Numbers having a DAM of 7 (see A061424): 7, 59, 68, 77, 86, 95, 399, 489, 498, 579, 588, 597, 669, 678, 687, 696, 759, 768, 777, 786, 795, 849, 858, 867, 876, 885, 894, 939, 948, 957, 966, 975, 984, 993, 1999, 2899, 2989, 2998, 3799, 3889, 3898, 3979, 3988, 3997, 4699, 4789, 4798, 4879, 4888, 4897, ...
Numbers having a DAM of 8 (see A061425): 8, 79, 88, 97, 699, 789, 798, 879, 888, 897, 969, 978, 987, 996, 5999, 6899, 6989, 6998, 7799, 7889, 7898, 7979, 7988, 7997, 8699, 8789, 8798, 8879, 8888, 8897, 8969, 8978, 8987, 8996, 9599, 9689, 9698, 9779, 9788, 9797, 9869, 9878, 9887, 9896, 9959, ...
Numbers having a DAM of 9 (see A002283): 9, 99, 999, 9999, 99999, 999999, 9999999, 99999999, 999999999, 9999999999, 99999999999, 999999999999, 9999999999999, 99999999999999, 999999999999999, 9999999999999999, 99999999999999999, 999999999999999999, 9999999999999999999, …

The Boomerang Rule
Let us assign to numbers whose DAM is not an integer a new DAM which is worth 0. Then let us replace one by one each digit "d" of the sequence S by the smallest number absent from S of which "d" is the DAM – and follow each result with a comma.
Jean-Marc Falcoz was quick to compute the first terms:

S=1,2,3,4,5,6,7,8,9,11,20,13,10,102,15,111,12,120,14,22,201,19,210,300,1003,1012,31,1021,40,16,1030,17,105,114,123,18,1102,1111,9,132,1120,21,24,23,25,1201,27,29,33,1210,30,1300,141,42,2002,2011,32,150,2020,26,34,2101,39,2110,36,51,38,2200,59,3001,41,28,3010,3100,35,4000,204,60,10004,79,10013,10022,43,213,10031,10040,10103,10112,99,10121,108,222,10130,10202,231,45,240,10211,303,44,312,117,321,37,10220,330,47,10301,402,68,411,999,126,135,10310,420,10400,49,144,50,11003,153,52,54,11012,53,11021,62,501,510,56,58,600,1007,61,11030,11102,162,1016,11111,46,63,1025,65,1034,67,1043,48,171,71,1052,11120,69,11201,180,9999,1061,11210,11300,70,207,57,55,12002,216,88,1070,1106,72,74,64,99999,225,76,78,12011,80,12020,1115,97,234,81,12101,83,243,12110,85,87,252,73,129,89,90,92,1124,94,138,66,96,12200,98,100,101,147,77,999999,13001,103,104,13010,261,13100,106,107,1133,1142,156,270,1151,14000,306,20003,109,110,315,20012,20021,112,113,165,115,20030,116,20102,118,324,20111,119,20120,20201,1160,9999999,99999999,20210,121,20300,1205,21002,21011,122,699,1214,1223,1232,21020,124,21101,333,125,21110,127,1241,128,1250,1304,342,21200,174,82,1313,183,130,22001,131,1322,22010,22100,351,133,360,192,219,405,23000,1331,30002,30011,86,414,1340,30020,423,95,30101,134,1403,1412,136,432,441,137,228,399,30110,139,450,140,30200,237,142,1421,75,789,246,31001,31010,999999999,9999999999,99999999999,31100,1430,84,32000,504,91,40001,143,513,40010,145,255,1502,146,40100,148,264,...


We see that the DAM line reproduces S (click on the above image to see it bigger). This is the boomerang effect – already met below the picture – merci and bravo Jean-Marc!






































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