Difference is a product of digits

Term a(n+1) of S is separated from a(n) by the difference “d”; “d” must be the product of a digit belonging to a(n) and a digit belonging to a(n+1).

Example:

S = 4, 12, 28, 34, 46, 74, 82, ...

from 4 to 12 there is 8 (that is 4x2)

from 12 to 28 there is 16 (that is 2x8)

from 28 to 34 there is 6 (that is 2x3)

from 34 to 46 there is 12 (that is 3x4)

from 46 to 74 there is 28 (that is 4x7)

from 74 to 82 there is 8 (that is 4x2)

...

One has sometimes multiple choices for a(n+1):

135, 136 (diff = 1x1)

135, 138 (diff = 1x3)

135, 147 (diff = 3x4)

135, 150 (diff = 3x5)

135, 165 (diff = 5x6)

135, 170 (diff = 5x7)

I’ve explored this together with Jean-Marc Falcoz at the beginning of March 2014.

Here are some results.

For S starting with 2, Jean-Marc has computed two sequences: [Sslow] which is the lexicographically slowest increasing one and [Sfast] which is the lexicographically fastest increasing one. An idea of their respective growing rates can be evaluated here:

[Sslow] = 2, 18, 19, 21, 23, 27, 33, 45, 65, 71, 78, 92, 101, 102, 103, 104, 105, 106, 107, ...

[Sfast] = 2, 18, 90, 135, 170, 205, 215, 230, 245, 290, 335, 370, 405, 425, 450, 495, 540, 580, 628, ...

Here are the first 500 terms of [Sslow]:

2, 18, 19, 21, 23, 27, 33, 45, 65, 71, 78, 92, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 201, 203, 207, 211, 212, 213, 214, 215, 216, 217, 218, 219, 221, 223, 227, 231, 233, 237, 241, 243, 247, 251, 253, 257, 261, 263, 267, 271, 273, 277, 281, 283, 287, 291, 293, 297, 303, 312, 313, 314, 315, 316, 317, 318, 319, 321, 323, 327, 333, 342, 348, 351, 354, 362, 368, 371, 374, 381, 384, 392, 398, 401, 405, 410, 411, 412, 413, 414, 415, 416, 417, 418, 419, 421, 423, 427, 431, 434, 442, 450, 466, 482, 490, 510, 511, 512, 513, 514, 515, 516, 517, 518, 519, 521, 523, 527, 533, 545, 565, 571, 576, 581, 586, 591, 596, 601, 607, 613, 614, 615, 616, 617, 618, 619, 621, 623, 627, 633, 645, 651, 656, 661, 667, 709, 716, 717, 718, 719, 721, 723, 727, 733, 745, 765, 771, 778, 792, 801, 809, 817, 818, 819, 821, 823, 827, 833, 841, 845, 865, 871, 878, 892, 901, 910, 911, 912, 913, 914, 915, 916, 917, 918, 919, 921, 923, 927, 933, 945, 965, 971, 978, 992, 1001, 1002, 1003, 1004, 1005, 1006, 1007, 1008, 1009, 1010, 1011, 1012, 1013, 1014, 1015, 1016, 1017, 1018, 1019, 1020, 1021, 1022, 1023, 1024, 1025, 1026, 1027, 1028, 1029, 1030, 1031, 1032, 1033, 1034, 1035, 1036, 1037, 1038, 1039, 1040, 1041, 1042, 1043, 1044, 1045, 1046, 1047, 1048, 1049, 1050, 1051, 1052, 1053, 1054, 1055, 1056, 1057, 1058, 1059, 1060, 1061, 1062, 1063, 1064, 1065, 1066, 1067, 1068, 1069, 1070, 1071, 1072, 1073, 1074, 1075, 1076, 1077, 1078, 1079, 1080, 1081, 1082, 1083, 1084, 1085, 1086, 1087, 1088, 1089, 1090, 1091, 1092, 1093, 1094, 1095, 1096, 1097, 1098, 1099, 1100, 1101, 1102, 1103, 1104, 1105, 1106, 1107, 1108, 1109, 1110, 1111, 1112, 1113, 1114, 1115, 1116, 1117, 1118, 1119, 1120, 1121, 1122, 1123, 1124, 1125, 1126, 1127, 1128, 1129, 1130, 1131, 1132, 1133, 1134, 1135, 1136, 1137, 1138, 1139, 1140, 1141, 1142, 1143, 1144, 1145, 1146, 1147, 1148, 1149, 1150, 1151, 1152, 1153, 1154, 1155, 1156, 1157, 1158, 1159, 1160, 1161, 1162, 1163, 1164, 1165, 1166, 1167, 1168, 1169, 1170, 1171, 1172, 1173, 1174, 1175, 1176, 1177, 1178, 1179, 1180, 1181, 1182, 1183, 1184, 1185, 1186, 1187, 1188, 1189, 1190, 1191, 1192, 1193, 1194, 1195, 1196, 1197, 1198, 1199, 1200, 1201, 1202, 1203, 1204, 1205, 1206, 1207, 1208, 1209, 1210, 1211, 1212, 1213, 1214, 1215, 1216, 1217, ...

[Sslow] — 500 terms

[Sslow] — 10000 terms

Here are the first 500 terms of [Sfast]

2, 18, 90, 135, 170, 205, 215, 230, 245, 290, 335, 370, 405, 425, 450, 495, 540, 580, 628, 676, 739, 811, 875, 947, 983, 991, 1000, 1009, 1090, 1135, 1170, 1205, 1215, 1230, 1245, 1290, 1335, 1370, 1405, 1425, 1450, 1495, 1540, 1580, 1628, 1676, 1739, 1811, 1875, 1947, 1983, 2001, 2019, 2037, 2052, 2097, 2115, 2130, 2145, 2190, 2235, 2270, 2305, 2320, 2335, 2370, 2405, 2425, 2450, 2495, 2540, 2580, 2628, 2676, 2739, 2811, 2875, 2947, 2983, 3010, 3025, 3050, 3095, 3122, 3149, 3181, 3237, 3258, 3306, 3327, 3390, 3435, 3470, 3505, 3530, 3560, 3608, 3656, 3692, 3755, 3811, 3875, 3947, 3983, 4019, 4055, 4091, 4127, 4190, 4235, 4270, 4305, 4325, 4350, 4395, 4431, 4451, 4496, 4568, 4616, 4652, 4697, 4760, 4816, 4880, 4952, 5006, 5036, 5090, 5135, 5170, 5205, 5230, 5260, 5308, 5356, 5386, 5426, 5462, 5498, 5579, 5633, 5681, 5737, 5786, 5850, 5922, 5976, 6048, 6096, 6168, 6216, 6252, 6297, 6351, 6396, 6468, 6516, 6552, 6597, 6651, 6696, 6768, 6832, 6904, 6985, 7048, 7104, 7153, 7202, 7258, 7314, 7363, 7412, 7468, 7524, 7580, 7636, 7699, 7762, 7818, 7890, 7971, 8043, 8107, 8179, 8251, 8315, 8379, 8451, 8515, 8579, 8651, 8715, 8779, 8851, 8923, 9004, 9085, 9166, 9247, 9328, 9409, 9490, 9571, 9652, 9733, 9814, 9895, 9976, 10048, 10096, 10168, 10216, 10240, 10264, 10280, 10304, 10328, 10376, 10439, 10475, 10510, 10535, 10570, 10612, 10648, 10704, 10753, 10809, 10890, 10971, 10980, 11025, 11050, 11095, 11104, 11112, 11128, 11176, 11239, 11257, 11271, 11289, 11316, 11340, 11364, 11391, 11427, 11490, 11535, 11570, 11612, 11648, 11704, 11753, 11809, 11890, 11971, 11980, 12025, 12050, 12095, 12113, 12127, 12190, 12235, 12270, 12305, 12320, 12335, 12370, 12405, 12425, 12450, 12495, 12540, 12580, 12628, 12676, 12739, 12811, 12875, 12947, 12983, 13010, 13025, 13050, 13095, 13122, 13149, 13181, 13237, 13258, 13306, 13327, 13390, 13435, 13470, 13505, 13530, 13560, 13608, 13656, 13692, 13755, 13811, 13875, 13947, 13983, 14019, 14055, 14091, 14127, 14190, 14235, 14270, 14305, 14325, 14350, 14395, 14431, 14451, 14496, 14568, 14616, 14652, 14697, 14760, 14816, 14880, 14952, 15006, 15036, 15090, 15135, 15170, 15205, 15230, 15260, 15308, 15356, 15386, 15426, 15462, 15498, 15579, 15633, 15681, 15737, 15786, 15850, 15922, 15976, 16048, 16096, 16168, 16216, 16252, 16297, 16351, 16396, 16468, 16516, 16552, 16597, 16651, 16696, 16768, 16832, 16904, 16985, 17048, 17104, 17153, 17202, 17258, 17314, 17363, 17412, 17468, 17524, 17580, 17636, 17699, 17762, 17818, 17890, 17971, 18043, 18107, 18179, 18251, 18315, 18379, 18451, 18515, 18579, 18651, 18715, 18779, 18851, 18923, 19004, 19085, 19166, 19247, 19328, 19409, 19490, 19571, 19652, 19733, 19814, 19895, 19976, 20048, 20096, 20168, 20216, 20240, 20264, 20280, 20304, 20328, 20376, 20439, 20475, 20510, 20535, 20570, 20612, 20648, 20704, 20753, 20809, 20890, 20971, 20989, 21013, 21027, 21090, 21135, 21170, 21205, 21215, 21230, 21245, 21290, 21335, 21370, 21405, 21425, 21450, 21495, 21540, 21580, 21628, 21676, 21739, 21811, 21875, 21947, 21983, 22001, 22019, 22037, 22052, 22097, 22115, 22130, 22145, 22190, 22235, 22270, 22305, 22320, 22335, 22370, 22405, 22425, 22450, 22495, 22540, 22580, 22628, 22676, 22739, 22811, 22875, 22947, 22983, 23010, 23025, 23050, 23095, 23122, 23149, 23181, 23237, ...

[Sfast] — 100 terms

[Sfast] — 10000 terms

Some integers have no successors – they are:

[Nosuccessor] = 0, 1, 3, 55, 57, 59, 66, 67, 69, 73, 75, 77, 79, 80, 955, 957, 959, 967, 969, 973, 975, 977, 979, 9955, 9957, 9959, 9967, 9969, 9973, 9975, 9977, 9979, 99955, 99957, 99959, 99967, 99969, 99973, 99975, 99977, 99979, ...

This sequence is infinite and shows from there on the repeated pattern:

(9....9)55

(9....9)57

(9....9)59

(9....9)67

(9....9)69

(9....9)73

(9....9)75

(9....9)77

(9....9)79

Some integers have no predecessors – the list seems to be finite:

[Nopredecessor] = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 22, 26, 55, 60, 63, 94.

Here is a “4-level” graph showing which integer goes where:

A “5-level” graph:

A “6-level” (zoomable) graph:

Many thanks to Jean-Marc!

Those questions remain open:

- what happens if we drop the “monotonically increasing” constraint? We would then allow a(n) > a(n+1);

- what about jumping from a(n) to a(n+1) only if the difference between the two terms is the product of their two biggest digits? (or the product of the two smallest ones?)

Best,

É.