Digits and gaps

NAME

Two terms that contain the digit « d » are always separated by « d » terms that do not contain the digit « d ». This is the lexicographically earliest sequence S of distinct nonnegative integers with this property.

DATA (only the first 9 terms are lexico-correct – did notice Jean-Marc Falcoz the next day): 

S = 0, 10, 20, 100, 30, 102, 40, 101, 203, 110, 50, 1024, 300, 1000, 200, 1001, 3045, 120, 60, 1010, 230, 104, 500, 201, 303, 106, 204, 1011, 305, 210, 70, 140, 2036, 1100, 505, 1002, 304, 1101, 207, 160, 350, 1042, 80, 1110, 302, 10000, 40567, 1020, 330, 10001, 202, 1048, 503, 1026, 700, 10010, 2034, 10011, 550, 1022, 3068, 401, 270, 10100, 530, 1021, 400, 601, 90, 108,...

EXAMPLE

As we start S with a(1) = 0, the digit 0 must be present in every term of the sequence. 

S must be extended now with a(2) = 10 as 10 is the smallest integer not present in S that contains the digit 0.

The next term will be a(3) = 20 as 20 is the smallest integer not present in S that contains the digit 0.

The next term will be a(4) = 100 as 100 is the smallest integer not present in S that contains both the digits 0 and 1.

The next term will be a(5) = 30 as 30 is the smallest integer not present in S that contains the digit 0.

The next term will be a(6) = 102 as 102 is the smallest integer not present in S that contains the digits 0, 1 and 2.

The next term will be a(7) = 40 as 40 is the smallest integer not present in S that contains the digit 0.

The next term will be a(8) = 101 as 101 is the smallest integer not present in S that contains both the digits 0 and 1.

Etc.

COMMENTS

Will the sequence S stop at some point?

___________________________

Update #1, Aug 6th, 2024 – Jean-Marc Falcoz

> Je crois qu’effectivement S est finie (indeed, the seq is finite – there is no 55th term)

S ={0,10,20,100,30,102,40,101,203,105,60,1024,300,110,200,150,304,1026,70,1000,230,1045,80,120,306,1001,2047,501,303,201,90,10468,302,510,700,210,340,1010,206,1005,3089,1042,707,1011,320,1056,400,1002,330,108,2079,1054,360,1012}

____________________

Update #2, Michael S. Branicky

MSB

Instead of 0,10,20,100,30,102,40,101,203,105,60,1024,300,110,... I get 0,10,20,100,30,102,40,101,203,105,60,1024,300,107,...

ÉA

You are right, Michael, thank you!

This is now A375232 – a warm merci to Jean-Marc and Michael!

____________________

Update #3, Aug 7th, 2024

What about extending the sequence with -1 when no more terms are available? And proceed from there as before?

Jean-Marc was quick to use this technique, extend the sequence to 1000 terms and compute a graph:

S ={0,10,20,100,30,102,40,101,203,105,60,1024,300,107,200,150,304,1026,80,109,230,10457,-1,120,306,110,204,1058,303,10279,-1,1046,302,501,-1,201,3048,170,206,1059,330,1042,-1,1000,320,105678,400,210,3000,190,202,1045,360,1027,800,1001,2034,510,-1,10269,3003,1047,220,1085,3030,1002,406,1010,2003,10579,-1,1204,308,106,2000,1005,340,1072,-1,901,2036,10458,-1,1012,3033,701,240,1056,3300,1029,808,104,2023,1057,600,1020,403,1011,2002,10589,3303,102467,-1,1100,2030,1015,404,1021,3068,1079,2020,1054,3330,1022,-1,160,2043,10578,-1,1092,30000,140,260,1050,30003,1207,408,1101,2032,10569,-1,1240,30030,710,2022,1508,3046,1102,-1,910,2033,10475,-1,1062,380,1110,402,1051,30033,10297,606,401,2203,1580,-1,1120,430,1067,2200,1095,30300,1402,880,10000,2063,1075,440,1200,30303,1009,2202,104568,30330,1270,-1,10001,2304,1055,660,1209,803,1074,2220,1105,30333,1206,4000,10010,2230,105789,-1,1420,603,10011,20000,1150,3004,1702,8000,1069,2300,1405,-1,1201,33000,1007,2046,1805,33003,1290,-1,410,2302,10567,-1,1202,3084,10100,20002,1509,630,10247,-1,10101,2303,1850,4004,1260,33030,1097,20020,1450,33033,1210,608,10110,2340,1507,-1,1902,33300,1064,20022,5018,33303,1720,4040,10111,2306,1590,-1,2014,830,1017,20200,1065,3034,1220,-1,1019,2320,104578,6000,2001,33330,11000,420,1500,300000,102679,8008,1004,2330,1501,-1,2010,3064,1070,20202,10598,300003,2041,-1,601,3002,1570,4044,2011,3008,1090,602,1504,300030,2017,-1,11001,2403,10568,-1,1920,300033,1407,20220,1505,3006,2012,480,11010,3020,10597,-1,10246,300300,11011,20222,5081,3040,2071,6006,1091,3022,1540,-1,2021,3038,1076,2004,1510,300303,2019,-1,1014,2360,10587,-1,2100,3043,11100,22000,10596,300330,10274,8080,11101,3023,1550,460,2101,300333,1709,22002,10485,303000,1602,-1,11110,2430,1705,-1,2091,3086,1040,22020,5001,303003,2107,4400,610,3032,10859,-1,2104,303030,1071,620,5010,3044,2102,8088,1099,3200,104567,-1,2110,303033,100000,2024,5108,3036,10729,-1,1041,3202,5011,-1,1620,3408,1077,22022,1905,303300,2140,6060,100001,3203,10758,4404,2120,303303,1096,22200,4015,303330,2170,8800,100010,20346,5015,-1,2109,303333,1470,22202,10586,330000,2201,4440,100011,3220,10759,6066,2401,3080,100100,22220,5051,3304,10267,-1,1109,3230,10548,-1,2210,3060,1107,2040,5100,330003,2190,8808,1406,3302,1750,-1,10002,3340,100101,2006,10895,330030,10427,-1,100110,3320,1506,40000,10012,3083,1790,200000,4051,3063,10020,-1,100111,3024,10785,-1,10296,330033,1044,200002,5101,330300,2701,4068,101000,20003,1950,-1,2410,330303,1607,200020,5180,3400,10021,-1,1190,2603,10547,-1,10022,3088,101001,2042,1560,330330,10792,-1,1104,20023,5801,6600,10102,3403,1170,200022,5019,330333,10264,8880,101010,20030,5017,40004,10112,3066,1900,200200,10584,333000,2710,-1,1006,3042,5105,-1,2901,3308,1704,2026,5110,333003,10120,40040,101011,20032,-1,4012,333030,101100,200202,5150,3406,7012,80000,1901,20033,4105,-1,2016,333033,1700,2044,5810,333300,2910,6606,1140,20203,5071,-1,10121,3480,1016,200220,5091,333303,10472,-1,101101,2630,8015,40044,10122,333330,1907,200222,10456,10200,80008,101110,3204,5107,6660,9012,1400,202000,8051,10276,40400,101111,20223,5109,-1,4021,3608,1701,202002,5501,3404,10201,-1,1609,20230,104587,-1,10202,110000,2064,5510,10927,80080,1401,20232,1605,-1,10210,3430,1707,202020,10958,3306,4102,-1,110001,20233,5170,40404,2061,3380,1909,202022,4150,7021,60000,110010,3240,8105,-1,9021,10467,202200,10005,10211,804,110011,3026,10795,-1,4120,110100,202202,10658,3440,7102,-1,1910,20300,4501,60006,10212,3800,1710,2204,10015,10629,-1,1404,20302,10857,-1,10220,3460,110101,202220,5190,10724,80088,1060,20303,10050,40440,10221,1970,2060,10845,10222,-1,110110,3402,10576,-1,9102,3803,1410,202222,10051,3360,7120,40444,110111,20320,10985,-1,10426,1770,220000,10055,4003,11002,680,1990,20322,10574,-1,11012,1061,2240,8150,10972,-1,1440,3062,10105,-1,11020,3804,7001,220002,10659,4201,-1,111000,20323,10875,604,11021,9001,220020,4510,10627,80800,111001,3420,10115,-1,9120,3600,1740,220022,8501,11022,44000,1066,20330,10957,-1,4210,3808,111010,2062,10150,4030,7201,-1,9010,20332,104586,-1,11102,7010,2400,10151,3603,9201,80808,4001,20333,5701,-1,2106,4033,111011,220200,15089,10742,60060,111100,22003,10155,44004,11120,3830,10679,220202,5014,11200,-1,111101,20364,15078,-1,9210,4010,220220,1650,7210,840,111110,22023,5901,60066,10024,7011,220222,8510,4034,2160,-1,9011,22030,10745,-1,11201,3680,1000000,2402,10500,12079,-1,1460,22032,10058,-1,11202,4043,7017,2066,5910,10042,80880,22033,10657,44040,11210,9019,222000,10854,3606,10027,-1,4023,10501,-1,10692,3880,4017,222002,10505,11220,640,22203,105798,-1,10124,1106,222020,10510,4300,10072,80888,9091,3206,5041,-1,12000,7071,2404,10685,10029,-1,4011,22230,5710,60600,12001,3840,222022,9015,102476,-1,22300,10085,44044,12002,3630,7019,222200,5104,12010,88000,1160,4032,7015,-1,10092,4014,2206,10158,10127,44400,22302,10695,-1,10142,8003,7100,222202,10511,3604,12011,-1,9100,22303,104758,-1,2601,2420,10515,12097,806,4041,22320,10550,-1,12012,4303,1670,222220,15098,10204,-1,3260,7051,44404,12020,8030,9101}

This variant has been submitted to the OEIS too (here).