2014 wishes – with jumps

From: Eric Angelini

Sent: Thursday, December 19, 2013 7:02 PM

To: Sequence Fanatics Discussion list

Subject: [seqfan] 2014 wishes -- with jumps

The idea behind the monotonically increasing sequence S is that the difference between a(n) and a(n+1) is given by the product z*a, "z" being the last digit of a(n) and "a" being the first digit of a(n+1).

An example of such a pair is (36,78); we see that |36-78|=42 and 42 is indeed 6*7.

My intuition is that S, whatever a(1), will always be finite. But, paradoxically, that the length of S can be arbitrarily long.

The finiteness of S comes from two facts:

- S stops immediately when "z", the last digit of a(n), is equal to 0;

- S cannot be extended, sometimes, because there is no product available to jump from a(n) to a(n+1).

An example is given by 58: if you try to start a(n+1) with "a" = 1 or 2 or 3 or... 9, you will fail.

For a(1)=6, for instance, S stops after 6,12,14,18,58.

Please note that a(n) can be followed, sometimes, by more than one a(n+1).

S will stop also after 6,12,14,18,66.

My question is:

What is the smallest a(1) producing a 2014-term sequence?

Best wishes to Neil, his team and all Seqfans!

;-D

Best,

É.

__________

[Lars Blomberg was (very quick) to answer]

> Starting with 18257, I find the following 2014 terms.

Regards

Lars

18257, 18264, 18268, 18276, 18282, 18284, 18288, 18296, 18302, 18304, 18308, 18316, 18322, 18324, 18328, 18336, 18342, 18344, 18348, 18356, 18362, 18364, 18368, 18376, 18382, 18384, 18388, 18396, 18402, 18404, 18408, 18416, 18422, 18424, 18428, 18436, 18442, 18444, 18448, 18456, 18462, 18464, 18468, 18476, 18482, 18484, 18488, 18496, 18502, 18504, 18508, 18516, 18522, 18524, 18528, 18536, 18542, 18544, 18548, 18556, 18562, 18564, 18568, 18576, 18582, 18584, 18588, 18596, 18602, 18604, 18608, 18616, 18622, 18624, 18628, 18636, 18642, 18644, 18648, 18656, 18662, 18664, 18668, 18676, 18682, 18684, 18688, 18696, 18702, 18704, 18708, 18716, 18722, 18724, 18728, 18736, 18742, 18744, 18748, 18756, 18762, 18764, 18768, 18776, 18782, 18784, 18788, 18796, 18802, 18804, 18808, 18816, 18822, 18824, 18828, 18836, 18842, 18844, 18848, 18856, 18862, 18864, 18868, 18876, 18882, 18884, 18888, 18896, 18902, 18904, 18908, 18916, 18922, 18924, 18928, 18936, 18942, 18944, 18948, 18956, 18962, 18964, 18968, 18976, 18982, 18984, 18988, 18996, 19002, 19004, 19008, 19016, 19022, 19024, 19028, 19036, 19042, 19044, 19048, 19056, 19062, 19064, 19068, 19076, 19082, 19084, 19088, 19096, 19102, 19104, 19108, 19116, 19122, 19124, 19128, 19136, 19142, 19144, 19148, 19156, 19162, 19164, 19168, 19176, 19182, 19184, 19188, 19196, 19202, 19204, 19208, 19216, 19222, 19224, 19228, 19236, 19242, 19244, 19248, 19256, 19262, 19264, 19268, 19276, 19282, 19284, 19288, 19296, 19302, 19304, 19308, 19316, 19322, 19324, 19328, 19336, 19342, 19344, 19348, 19356, 19362, 19364, 19368, 19376, 19382, 19384, 19388, 19396, 19402, 19404, 19408, 19416, 19422, 19424, 19428, 19436, 19442, 19444, 19448, 19456, 19462, 19464, 19468, 19476, 19482, 19484, 19488, 19496, 19502, 19504, 19508, 19516, 19522, 19524, 19528, 19536, 19542, 19544, 19548, 19556, 19562, 19564, 19568, 19576, 19582, 19584, 19588, 19596, 19602, 19604, 19608, 19616, 19622, 19624, 19628, 19636, 19642, 19644, 19648, 19656, 19662, 19664, 19668, 19676, 19682, 19684, 19688, 19696, 19702, 19704, 19708, 19716, 19722, 19724, 19728, 19736, 19742, 19744, 19748, 19756, 19762, 19764, 19768, 19776, 19782, 19784, 19788, 19796, 19802, 19804, 19808, 19816, 19822, 19824, 19828, 19836, 19842, 19844, 19848, 19856, 19862, 19864, 19868, 19876, 19882, 19884, 19888, 19896, 19902, 19904, 19908, 19916, 19922, 19924, 19928, 19936, 19942, 19944, 19948, 19956, 19962, 19964, 19968, 19976, 19982, 19984, 19988, 20004, 20012, 20016, 20028, 20044, 20052, 20056, 20068, 20084, 20092, 20096, 20108, 20124, 20132, 20136, 20148, 20164, 20172, 20176, 20188, 20204, 20212, 20216, 20228, 20244, 20252, 20256, 20268, 20284, 20292, 20296, 20308, 20324, 20332, 20336, 20348, 20364, 20372, 20376, 20388, 20404, 20412, 20416, 20428, 20444, 20452, 20456, 20468, 20484, 20492, 20496, 20508, 20524, 20532, 20536, 20548, 20564, 20572, 20576, 20588, 20604, 20612, 20616, 20628, 20644, 20652, 20656, 20668, 20684, 20692, 20696, 20708, 20724, 20732, 20736, 20748, 20764, 20772, 20776, 20788, 20804, 20812, 20816, 20828, 20844, 20852, 20856, 20868, 20884, 20892, 20896, 20908, 20924, 20932, 20936, 20948, 20964, 20972, 20976, 20988, 21004, 21012, 21016, 21028, 21044, 21052, 21056, 21068, 21084, 21092, 21096, 21108, 21124, 21132, 21136, 21148, 21164, 21172, 21176, 21188, 21204, 21212, 21216, 21228, 21244, 21252, 21256, 21268, 21284, 21292, 21296, 21308, 21324, 21332, 21336, 21348, 21364, 21372, 21376, 21388, 21404, 21412, 21416, 21428, 21444, 21452, 21456, 21468, 21484, 21492, 21496, 21508, 21524, 21532, 21536, 21548, 21564, 21572, 21576, 21588, 21604, 21612, 21616, 21628, 21644, 21652, 21656, 21668, 21684, 21692, 21696, 21708, 21724, 21732, 21736, 21748, 21764, 21772, 21776, 21788, 21804, 21812, 21816, 21828, 21844, 21852, 21856, 21868, 21884, 21892, 21896, 21908, 21924, 21932, 21936, 21948, 21964, 21972, 21976, 21988, 22004, 22012, 22016, 22028, 22044, 22052, 22056, 22068, 22084, 22092, 22096, 22108, 22124, 22132, 22136, 22148, 22164, 22172, 22176, 22188, 22204, 22212, 22216, 22228, 22244, 22252, 22256, 22268, 22284, 22292, 22296, 22308, 22324, 22332, 22336, 22348, 22364, 22372, 22376, 22388, 22404, 22412, 22416, 22428, 22444, 22452, 22456, 22468, 22484, 22492, 22496, 22508, 22524, 22532, 22536, 22548, 22564, 22572, 22576, 22588, 22604, 22612, 22616, 22628, 22644, 22652, 22656, 22668, 22684, 22692, 22696, 22708, 22724, 22732, 22736, 22748, 22764, 22772, 22776, 22788, 22804, 22812, 22816, 22828, 22844, 22852, 22856, 22868, 22884, 22892, 22896, 22908, 22924, 22932, 22936, 22948, 22964, 22972, 22976, 22988, 23004, 23012, 23016, 23028, 23044, 23052, 23056, 23068, 23084, 23092, 23096, 23108, 23124, 23132, 23136, 23148, 23164, 23172, 23176, 23188, 23204, 23212, 23216, 23228, 23244, 23252, 23256, 23268, 23284, 23292, 23296, 23308, 23324, 23332, 23336, 23348, 23364, 23372, 23376, 23388, 23404, 23412, 23416, 23428, 23444, 23452, 23456, 23468, 23484, 23492, 23496, 23508, 23524, 23532, 23536, 23548, 23564, 23572, 23576, 23588, 23604, 23612, 23616, 23628, 23644, 23652, 23656, 23668, 23684, 23692, 23696, 23708, 23724, 23732, 23736, 23748, 23764, 23772, 23776, 23788, 23804, 23812, 23816, 23828, 23844, 23852, 23856, 23868, 23884, 23892, 23896, 23908, 23924, 23932, 23936, 23948, 23964, 23972, 23976, 23988, 24004, 24012, 24016, 24028, 24044, 24052, 24056, 24068, 24084, 24092, 24096, 24108, 24124, 24132, 24136, 24148, 24164, 24172, 24176, 24188, 24204, 24212, 24216, 24228, 24244, 24252, 24256, 24268, 24284, 24292, 24296, 24308, 24324, 24332, 24336, 24348, 24364, 24372, 24376, 24388, 24404, 24412, 24416, 24428, 24444, 24452, 24456, 24468, 24484, 24492, 24496, 24508, 24524, 24532, 24536, 24548, 24564, 24572, 24576, 24588, 24604, 24612, 24616, 24628, 24644, 24652, 24656, 24668, 24684, 24692, 24696, 24708, 24724, 24732, 24736, 24748, 24764, 24772, 24776, 24788, 24804, 24812, 24816, 24828, 24844, 24852, 24856, 24868, 24884, 24892, 24896, 24908, 24924, 24932, 24936, 24948, 24964, 24972, 24976, 24988, 25004, 25012, 25016, 25028, 25044, 25052, 25056, 25068, 25084, 25092, 25096, 25108, 25124, 25132, 25136, 25148, 25164, 25172, 25176, 25188, 25204, 25212, 25216, 25228, 25244, 25252, 25256, 25268, 25284, 25292, 25296, 25308, 25324, 25332, 25336, 25348, 25364, 25372, 25376, 25388, 25404, 25412, 25416, 25428, 25444, 25452, 25456, 25468, 25484, 25492, 25496, 25508, 25524, 25532, 25536, 25548, 25564, 25572, 25576, 25588, 25604, 25612, 25616, 25628, 25644, 25652, 25656, 25668, 25684, 25692, 25696, 25708, 25724, 25732, 25736, 25748, 25764, 25772, 25776, 25788, 25804, 25812, 25816, 25828, 25844, 25852, 25856, 25868, 25884, 25892, 25896, 25908, 25924, 25932, 25936, 25948, 25964, 25972, 25976, 25988, 26004, 26012, 26016, 26028, 26044, 26052, 26056, 26068, 26084, 26092, 26096, 26108, 26124, 26132, 26136, 26148, 26164, 26172, 26176, 26188, 26204, 26212, 26216, 26228, 26244, 26252, 26256, 26268, 26284, 26292, 26296, 26308, 26324, 26332, 26336, 26348, 26364, 26372, 26376, 26388, 26404, 26412, 26416, 26428, 26444, 26452, 26456, 26468, 26484, 26492, 26496, 26508, 26524, 26532, 26536, 26548, 26564, 26572, 26576, 26588, 26604, 26612, 26616, 26628, 26644, 26652, 26656, 26668, 26684, 26692, 26696, 26708, 26724, 26732, 26736, 26748, 26764, 26772, 26776, 26788, 26804, 26812, 26816, 26828, 26844, 26852, 26856, 26868, 26884, 26892, 26896, 26908, 26924, 26932, 26936, 26948, 26964, 26972, 26976, 26988, 27004, 27012, 27016, 27028, 27044, 27052, 27056, 27068, 27084, 27092, 27096, 27108, 27124, 27132, 27136, 27148, 27164, 27172, 27176, 27188, 27204, 27212, 27216, 27228, 27244, 27252, 27256, 27268, 27284, 27292, 27296, 27308, 27324, 27332, 27336, 27348, 27364, 27372, 27376, 27388, 27404, 27412, 27416, 27428, 27444, 27452, 27456, 27468, 27484, 27492, 27496, 27508, 27524, 27532, 27536, 27548, 27564, 27572, 27576, 27588, 27604, 27612, 27616, 27628, 27644, 27652, 27656, 27668, 27684, 27692, 27696, 27708, 27724, 27732, 27736, 27748, 27764, 27772, 27776, 27788, 27804, 27812, 27816, 27828, 27844, 27852, 27856, 27868, 27884, 27892, 27896, 27908, 27924, 27932, 27936, 27948, 27964, 27972, 27976, 27988, 28004, 28012, 28016, 28028, 28044, 28052, 28056, 28068, 28084, 28092, 28096, 28108, 28124, 28132, 28136, 28148, 28164, 28172, 28176, 28188, 28204, 28212, 28216, 28228, 28244, 28252, 28256, 28268, 28284, 28292, 28296, 28308, 28324, 28332, 28336, 28348, 28364, 28372, 28376, 28388, 28404, 28412, 28416, 28428, 28444, 28452, 28456, 28468, 28484, 28492, 28496, 28508, 28524, 28532, 28536, 28548, 28564, 28572, 28576, 28588, 28604, 28612, 28616, 28628, 28644, 28652, 28656, 28668, 28684, 28692, 28696, 28708, 28724, 28732, 28736, 28748, 28764, 28772, 28776, 28788, 28804, 28812, 28816, 28828, 28844, 28852, 28856, 28868, 28884, 28892, 28896, 28908, 28924, 28932, 28936, 28948, 28964, 28972, 28976, 28988, 29004, 29012, 29016, 29028, 29044, 29052, 29056, 29068, 29084, 29092, 29096, 29108, 29124, 29132, 29136, 29148, 29164, 29172, 29176, 29188, 29204, 29212, 29216, 29228, 29244, 29252, 29256, 29268, 29284, 29292, 29296, 29308, 29324, 29332, 29336, 29348, 29364, 29372, 29376, 29388, 29404, 29412, 29416, 29428, 29444, 29452, 29456, 29468, 29484, 29492, 29496, 29508, 29524, 29532, 29536, 29548, 29564, 29572, 29576, 29588, 29604, 29612, 29616, 29628, 29644, 29652, 29656, 29668, 29684, 29692, 29696, 29708, 29724, 29732, 29736, 29748, 29764, 29772, 29776, 29788, 29804, 29812, 29816, 29828, 29844, 29852, 29856, 29868, 29884, 29892, 29896, 29908, 29924, 29932, 29936, 29948, 29964, 29972, 29976, 29988, 30012, 30018, 30042, 30048, 30072, 30078, 30102, 30108, 30132, 30138, 30162, 30168, 30192, 30198, 30222, 30228, 30252, 30258, 30282, 30288, 30312, 30318, 30342, 30348, 30372, 30378, 30402, 30408, 30432, 30438, 30462, 30468, 30492, 30498, 30522, 30528, 30552, 30558, 30582, 30588, 30612, 30618, 30642, 30648, 30672, 30678, 30702, 30708, 30732, 30738, 30762, 30768, 30792, 30798, 30822, 30828, 30852, 30858, 30882, 30888, 30912, 30918, 30942, 30948, 30972, 30978, 31002, 31008, 31032, 31038, 31062, 31068, 31092, 31098, 31122, 31128, 31152, 31158, 31182, 31188, 31212, 31218, 31242, 31248, 31272, 31278, 31302, 31308, 31332, 31338, 31362, 31368, 31392, 31398, 31422, 31428, 31452, 31458, 31482, 31488, 31512, 31518, 31542, 31548, 31572, 31578, 31602, 31608, 31632, 31638, 31662, 31668, 31692, 31698, 31722, 31728, 31752, 31758, 31782, 31788, 31812, 31818, 31842, 31848, 31872, 31878, 31902, 31908, 31932, 31938, 31962, 31968, 31992, 31998, 32022, 32028, 32052, 32058, 32082, 32088, 32112, 32118, 32142, 32148, 32172, 32178, 32202, 32208, 32232, 32238, 32262, 32268, 32292, 32298, 32322, 32328, 32352, 32358, 32382, 32388, 32412, 32418, 32442, 32448, 32472, 32478, 32502, 32508, 32532, 32538, 32562, 32568, 32592, 32598, 32622, 32628, 32652, 32658, 32682, 32688, 32712, 32718, 32742, 32748, 32772, 32778, 32802, 32808, 32832, 32838, 32862, 32868, 32892, 32898, 32922, 32928, 32952, 32958, 32982, 32988, 33012, 33018, 33042, 33048, 33072, 33078, 33102, 33108, 33132, 33138, 33162, 33168, 33192, 33198, 33222, 33228, 33252, 33258, 33282, 33288, 33312, 33318, 33342, 33348, 33372, 33378, 33402, 33408, 33432, 33438, 33462, 33468, 33492, 33498, 33522, 33528, 33552, 33558, 33582, 33588, 33612, 33618, 33642, 33648, 33672, 33678, 33702, 33708, 33732, 33738, 33762, 33768, 33792, 33798, 33822, 33828, 33852, 33858, 33882, 33888, 33912, 33918, 33942, 33948, 33972, 33978, 34002, 34008, 34032, 34038, 34062, 34068, 34092, 34098, 34122, 34128, 34152, 34158, 34182, 34188, 34212, 34218, 34242, 34248, 34272, 34278, 34302, 34308, 34332, 34338, 34362, 34368, 34392, 34398, 34422, 34428, 34452, 34458, 34482, 34488, 34512, 34518, 34542, 34548, 34572, 34578, 34602, 34608, 34632, 34638, 34662, 34668, 34692, 34698, 34722, 34728, 34752, 34758, 34782, 34788, 34812, 34818, 34842, 34848, 34872, 34878, 34902, 34908, 34932, 34938, 34962, 34968, 34992, 34998, 35022, 35028, 35052, 35058, 35082, 35088, 35112, 35118, 35142, 35148, 35172, 35178, 35202, 35208, 35232, 35238, 35262, 35268, 35292, 35298, 35322, 35328, 35352, 35358, 35382, 35388, 35412, 35418, 35442, 35448, 35472, 35478, 35502, 35508, 35532, 35538, 35562, 35568, 35592, 35598, 35622, 35628, 35652, 35658, 35682, 35688, 35712, 35718, 35742, 35748, 35772, 35778, 35802, 35808, 35832, 35838, 35862, 35868, 35892, 35898, 35922, 35928, 35952, 35958, 35982, 35988, 36012, 36018, 36042, 36048, 36072, 36078, 36102, 36108, 36132, 36138, 36162, 36168, 36192, 36198, 36222, 36228, 36252, 36258, 36282, 36288, 36312, 36318, 36342, 36348, 36372, 36378, 36402, 36408, 36432, 36438, 36462, 36468, 36492, 36498, 36522, 36528, 36552, 36558, 36582, 36588, 36612, 36618, 36642, 36648, 36672, 36678, 36702, 36708, 36732, 36738, 36762, 36768, 36792, 36798, 36822, 36828, 36852, 36858, 36882, 36888, 36912, 36918, 36942, 36948, 36972, 36978, 37002, 37008, 37032, 37038, 37062, 37068, 37092, 37098, 37122, 37128, 37152, 37158, 37182, 37188, 37212, 37218, 37242, 37248, 37272, 37278, 37302, 37308, 37332, 37338, 37362, 37368, 37392, 37398, 37422, 37428, 37452, 37458, 37482, 37488, 37512, 37518, 37542, 37548, 37572, 37578, 37602, 37608, 37632, 37638, 37662, 37668, 37692, 37698, 37722, 37728, 37752, 37758, 37782, 37788, 37812, 37818, 37842, 37848, 37872, 37878, 37902, 37908, 37932, 37938, 37962, 37968, 37992, 37998, 38022, 38028, 38052, 38058, 38082, 38088, 38112, 38118, 38142, 38148, 38172, 38178, 38202, 38208, 38232, 38238, 38262, 38268, 38292, 38298, 38322, 38328, 38352, 38358, 38382, 38388, 38412, 38418, 38442, 38448, 38472, 38478, 38502, 38508, 38532, 38538, 38562, 38568, 38592, 38598, 38622, 38628, 38652, 38658, 38682, 38688, 38712, 38718, 38742, 38748, 38772, 38778, 38802, 38808, 38832, 38838, 38862, 38868, 38892, 38898, 38922, 38928, 38952, 38958, 38982, 38988, 39012, 39018, 39042, 39048, 39072, 39078, 39102, 39108, 39132, 39138, 39162, 39168, 39192, 39198, 39222, 39228, 39252, 39258, 39282, 39288, 39312, 39318, 39342, 39348, 39372, 39378, 39402, 39408, 39432, 39438, 39462, 39468, 39492, 39498, 39522, 39528, 39552, 39558, 39582, 39588, 39612, 39618, 39642, 39648, 39672, 39678, 39702, 39708, 39732, 39738, 39762, 39768, 39792, 39798, 39822, 39828, 39852, 39858, 39882, 39888, 39912, 39918, 39942, 39948, 39972, 39978, 40010

[Me]:

> Geeeeeeee ! Lars !

[Lars, on December 22^nd, 2013]]:

Below are some more data about the sequence. Why numbers with initial digit = 4 is so abundant as last term, I don't know, but I will try to figure it out using my feeble mathematical skills.

Best regards,

Lars

Large solution counts (for start values up to 100000):

=====================================================

10 for start 7, 187

11 for start 8, 1987, 19987

23 for start 9 (the largest number of solutions for a single start value so far)

The 23 solutions for 9:

9,18,58

9,18,66

9,18,74

9,18,82

9,18,90

9,27,69

9,27,76

9,27,83

9,27,90

9,36,78

9,36,84

9,36,90

9,45,85

9,45,90

9,54,86

9,54,90

9,63,87

9,63,90

9,72,88

9,72,90

9,81,89

9,81,90

9,90

The 11 solutions for 8:

8,16,28

8,16,34,54,86

8,16,34,54,90

8,16,40

8,24,36,78

8,24,36,84

8,24,36,90

8,24,40

8,32,38

8,32,40

8,40

The 11 solutions for 1987:

1987, 1994, 1998, 2014, 2022, 2026, 2038, 2054, 2062, 2066, 2078, 2094, 2102, 2106, 2118, 2134, 2142, 2146, 2158, 2174, 2182, 2186, 2198, 2214, 2222, 2226, 2238, 2254, 2262, 2266, 2278, 2294, 2302, 2306, 2318, 2334, 2342, 2346, 2358, 2374, 2382, 2386, 2398, 2414, 2422, 2426, 2438, 2454, 2462, 2466, 2478, 2494, 2502, 2506, 2518, 2534, 2542, 2546, 2558, 2574, 2582, 2586, 2598, 2614, 2622, 2626, 2638, 2654, 2662, 2666, 2678, 2694, 2702, 2706, 2718, 2734, 2742, 2746, 2758, 2774, 2782, 2786, 2798, 2814, 2822, 2826, 2838, 2854, 2862, 2866, 2878, 2894, 2902, 2906, 2918, 2934, 2942, 2946, 2958, 2974, 2982, 2986, 2998, 3022, 3028, 3052, 3058, 3082, 3088, 3112, 3118, 3142, 3148, 3172, 3178, 3202, 3208, 3232, 3238, 3262, 3268, 3292, 3298, 3322, 3328, 3352, 3358, 3382, 3388, 3412, 3418, 3442, 3448, 3472, 3478, 3502, 3508, 3532, 3538, 3562, 3568, 3592, 3598, 3622, 3628, 3652, 3658, 3682, 3688, 3712, 3718, 3742, 3748, 3772, 3778, 3802, 3808, 3832, 3838, 3862, 3868, 3892, 3898, 3922, 3928, 3952, 3958, 3982, 3988, 4020,

1987, 1994, 1998, 2014, 2022, 2026, 2038, 2054, 2062, 2066, 2078, 2094, 2102, 2106, 2118, 2134, 2142, 2146, 2158, 2174, 2182, 2186, 2198, 2214, 2222, 2226, 2238, 2254, 2262, 2266, 2278, 2294, 2302, 2306, 2318, 2334, 2342, 2346, 2358, 2374, 2382, 2386, 2398, 2414, 2422, 2426, 2438, 2454, 2462, 2466, 2478, 2494, 2502, 2506, 2518, 2534, 2542, 2546, 2558, 2574, 2582, 2586, 2598, 2614, 2622, 2626, 2638, 2654, 2662, 2666, 2678, 2694, 2702, 2706, 2718, 2734, 2742, 2746, 2758, 2774, 2782, 2786, 2798, 2814, 2822, 2826, 2838, 2854, 2862, 2866, 2878, 2894, 2902, 2906, 2918, 2934, 2942, 2946, 2958, 2974, 2982, 2986, 3004, 3016, 3034, 3046, 3064, 3076, 3094, 3106, 3124, 3136, 3154, 3166, 3184, 3196, 3214, 3226, 3244, 3256, 3274, 3286, 3304, 3316, 3334, 3346, 3364, 3376, 3394, 3406, 3424, 3436, 3454, 3466, 3484, 3496, 3514, 3526, 3544, 3556, 3574, 3586, 3604, 3616, 3634, 3646, 3664, 3676, 3694, 3706, 3724, 3736, 3754, 3766, 3784, 3796, 3814, 3826, 3844, 3856, 3874, 3886, 3904, 3916, 3934, 3946, 3964, 3976, 3994, 4010,

1987, 1994, 1998, 2014, 2022, 2026, 2038, 2054, 2062, 2066, 2078, 2094, 2102, 2106, 2118, 2134, 2142, 2146, 2158, 2174, 2182, 2186, 2198, 2214, 2222, 2226, 2238, 2254, 2262, 2266, 2278, 2294, 2302, 2306, 2318, 2334, 2342, 2346, 2358, 2374, 2382, 2386, 2398, 2414, 2422, 2426, 2438, 2454, 2462, 2466, 2478, 2494, 2502, 2506, 2518, 2534, 2542, 2546, 2558, 2574, 2582, 2586, 2598, 2614, 2622, 2626, 2638, 2654, 2662, 2666, 2678, 2694, 2702, 2706, 2718, 2734, 2742, 2746, 2758, 2774, 2782, 2786, 2798, 2814, 2822, 2826, 2838, 2854, 2862, 2866, 2878, 2894, 2902, 2906, 2918, 2934, 2942, 2946, 2958, 2974, 2982, 2986, 3004, 3016, 3034, 3046, 3064, 3076, 3094, 3106, 3124, 3136, 3154, 3166, 3184, 3196, 3214, 3226, 3244, 3256, 3274, 3286, 3304, 3316, 3334, 3346, 3364, 3376, 3394, 3406, 3424, 3436, 3454, 3466, 3484, 3496, 3514, 3526, 3544, 3556, 3574, 3586, 3604, 3616, 3634, 3646, 3664, 3676, 3694, 3706, 3724, 3736, 3754, 3766, 3784, 3796, 3814, 3826, 3844, 3856, 3874, 3886, 3904, 3916, 3934, 3946, 3964, 3976, 4000,

1987, 1994, 2002, 2006, 2018, 2034, 2042, 2046, 2058, 2074, 2082, 2086, 2098, 2114, 2122, 2126, 2138, 2154, 2162, 2166, 2178, 2194, 2202, 2206, 2218, 2234, 2242, 2246, 2258, 2274, 2282, 2286, 2298, 2314, 2322, 2326, 2338, 2354, 2362, 2366, 2378, 2394, 2402, 2406, 2418, 2434, 2442, 2446, 2458, 2474, 2482, 2486, 2498, 2514, 2522, 2526, 2538, 2554, 2562, 2566, 2578, 2594, 2602, 2606, 2618, 2634, 2642, 2646, 2658, 2674, 2682, 2686, 2698, 2714, 2722, 2726, 2738, 2754, 2762, 2766, 2778, 2794, 2802, 2806, 2818, 2834, 2842, 2846, 2858, 2874, 2882, 2886, 2898, 2914, 2922, 2926, 2938, 2954, 2962, 2966, 2978, 2994, 3006, 3024, 3036, 3054, 3066, 3084, 3096, 3114, 3126, 3144, 3156, 3174, 3186, 3204, 3216, 3234, 3246, 3264, 3276, 3294, 3306, 3324, 3336, 3354, 3366, 3384, 3396, 3414, 3426, 3444, 3456, 3474, 3486, 3504, 3516, 3534, 3546, 3564, 3576, 3594, 3606, 3624, 3636, 3654, 3666, 3684, 3696, 3714, 3726, 3744, 3756, 3774, 3786, 3804, 3816, 3834, 3846, 3864, 3876, 3894, 3906, 3924, 3936, 3954, 3966, 3984, 3996, 4020,

1987, 1994, 2002, 2006, 2018, 2034, 2042, 2046, 2058, 2074, 2082, 2086, 2098, 2114, 2122, 2126, 2138, 2154, 2162, 2166, 2178, 2194, 2202, 2206, 2218, 2234, 2242, 2246, 2258, 2274, 2282, 2286, 2298, 2314, 2322, 2326, 2338, 2354, 2362, 2366, 2378, 2394, 2402, 2406, 2418, 2434, 2442, 2446, 2458, 2474, 2482, 2486, 2498, 2514, 2522, 2526, 2538, 2554, 2562, 2566, 2578, 2594, 2602, 2606, 2618, 2634, 2642, 2646, 2658, 2674, 2682, 2686, 2698, 2714, 2722, 2726, 2738, 2754, 2762, 2766, 2778, 2794, 2802, 2806, 2818, 2834, 2842, 2846, 2858, 2874, 2882, 2886, 2898, 2914, 2922, 2926, 2938, 2954, 2962, 2966, 2978, 3002, 3008, 3032, 3038, 3062, 3068, 3092, 3098, 3122, 3128, 3152, 3158, 3182, 3188, 3212, 3218, 3242, 3248, 3272, 3278, 3302, 3308, 3332, 3338, 3362, 3368, 3392, 3398, 3422, 3428, 3452, 3458, 3482, 3488, 3512, 3518, 3542, 3548, 3572, 3578, 3602, 3608, 3632, 3638, 3662, 3668, 3692, 3698, 3722, 3728, 3752, 3758, 3782, 3788, 3812, 3818, 3842, 3848, 3872, 3878, 3902, 3908, 3932, 3938, 3962, 3968, 3992, 3998, 4030,

1987, 1994, 2002, 2006, 2018, 2034, 2042, 2046, 2058, 2074, 2082, 2086, 2098, 2114, 2122, 2126, 2138, 2154, 2162, 2166, 2178, 2194, 2202, 2206, 2218, 2234, 2242, 2246, 2258, 2274, 2282, 2286, 2298, 2314, 2322, 2326, 2338, 2354, 2362, 2366, 2378, 2394, 2402, 2406, 2418, 2434, 2442, 2446, 2458, 2474, 2482, 2486, 2498, 2514, 2522, 2526, 2538, 2554, 2562, 2566, 2578, 2594, 2602, 2606, 2618, 2634, 2642, 2646, 2658, 2674, 2682, 2686, 2698, 2714, 2722, 2726, 2738, 2754, 2762, 2766, 2778, 2794, 2802, 2806, 2818, 2834, 2842, 2846, 2858, 2874, 2882, 2886, 2898, 2914, 2922, 2926, 2938, 2954, 2962, 2966, 2978, 3002, 3008, 3032, 3038, 3062, 3068, 3092, 3098, 3122, 3128, 3152, 3158, 3182, 3188, 3212, 3218, 3242, 3248, 3272, 3278, 3302, 3308, 3332, 3338, 3362, 3368, 3392, 3398, 3422, 3428, 3452, 3458, 3482, 3488, 3512, 3518, 3542, 3548, 3572, 3578, 3602, 3608, 3632, 3638, 3662, 3668, 3692, 3698, 3722, 3728, 3752, 3758, 3782, 3788, 3812, 3818, 3842, 3848, 3872, 3878, 3902, 3908, 3932, 3938, 3962, 3968, 3992, 4000,

1987, 1994, 2002, 2006, 2018, 2034, 2042, 2046, 2058, 2074, 2082, 2086, 2098, 2114, 2122, 2126, 2138, 2154, 2162, 2166, 2178, 2194, 2202, 2206, 2218, 2234, 2242, 2246, 2258, 2274, 2282, 2286, 2298, 2314, 2322, 2326, 2338, 2354, 2362, 2366, 2378, 2394, 2402, 2406, 2418, 2434, 2442, 2446, 2458, 2474, 2482, 2486, 2498, 2514, 2522, 2526, 2538, 2554, 2562, 2566, 2578, 2594, 2602, 2606, 2618, 2634, 2642, 2646, 2658, 2674, 2682, 2686, 2698, 2714, 2722, 2726, 2738, 2754, 2762, 2766, 2778, 2794, 2802, 2806, 2818, 2834, 2842, 2846, 2858, 2874, 2882, 2886, 2898, 2914, 2922, 2926, 2938, 2954, 2962, 2966, 2978, 3002, 3008, 3032, 3038, 3062, 3068, 3092, 3098, 3122, 3128, 3152, 3158, 3182, 3188, 3212, 3218, 3242, 3248, 3272, 3278, 3302, 3308, 3332, 3338, 3362, 3368, 3392, 3398, 3422, 3428, 3452, 3458, 3482, 3488, 3512, 3518, 3542, 3548, 3572, 3578, 3602, 3608, 3632, 3638, 3662, 3668, 3692, 3698, 3722, 3728, 3752, 3758, 3782, 3788, 3812, 3818, 3842, 3848, 3872, 3878, 3902, 3908, 3932, 3938, 3962, 3968, 4000,

1987, 2001, 2003, 2009, 2027, 2041, 2043, 2049, 2067, 2081, 2083, 2089, 2107, 2121, 2123, 2129, 2147, 2161, 2163, 2169, 2187, 2201, 2203, 2209, 2227, 2241, 2243, 2249, 2267, 2281, 2283, 2289, 2307, 2321, 2323, 2329, 2347, 2361, 2363, 2369, 2387, 2401, 2403, 2409, 2427, 2441, 2443, 2449, 2467, 2481, 2483, 2489, 2507, 2521, 2523, 2529, 2547, 2561, 2563, 2569, 2587, 2601, 2603, 2609, 2627, 2641, 2643, 2649, 2667, 2681, 2683, 2689, 2707, 2721, 2723, 2729, 2747, 2761, 2763, 2769, 2787, 2801, 2803, 2809, 2827, 2841, 2843, 2849, 2867, 2881, 2883, 2889, 2907, 2921, 2923, 2929, 2947, 2961, 2963, 2969, 2987, 3008, 3032, 3038, 3062, 3068, 3092, 3098, 3122, 3128, 3152, 3158, 3182, 3188, 3212, 3218, 3242, 3248, 3272, 3278, 3302, 3308, 3332, 3338, 3362, 3368, 3392, 3398, 3422, 3428, 3452, 3458, 3482, 3488, 3512, 3518, 3542, 3548, 3572, 3578, 3602, 3608, 3632, 3638, 3662, 3668, 3692, 3698, 3722, 3728, 3752, 3758, 3782, 3788, 3812, 3818, 3842, 3848, 3872, 3878, 3902, 3908, 3932, 3938, 3962, 3968, 3992, 3998, 4030,

Statistics of first digit of the last term when sequence stops:

==============================================================

For start values 1..10^6.

The sum is > 10^6 because there are sometimes several possibilities for the next number.

1 22225

2 11120

3 44448

4 892435

5 105549

6 11133

7 37064

8 11153

9 676105

Sum 1811232

There are remarkable differences in the frequencies.

Numbers with initial digit 4 are much more common as last term than 2 or 6.

Count=2014 solutions below 1000000

==================================

First Last

18257 40010

18258 40000

18259 40010

18262 40010

18263 40000

18264 40010

18266 40000

18267 40020

18267 40010

18267 40020

18267 40030

18271 40000

18272 40020

18272 40010

18272 40020

18272 40030

279865 300000

279875 300010

369788 400000

369789 400000

369796 400000

369797 400000

369798 400010

369799 400010

369803 400000

369804 400000

369806 400010

369808 400020

369811 400000

369812 400000

369813 400010

369814 400010

369816 400020

369818 400030

369821 400010

369823 400020

369831 400020

369833 400030

459719 500000

459727 500000

459729 500010

459735 500000

459737 500010

459739 500020

459743 500000

459745 500010

459747 500020

459751 500000

459753 500010

459755 500020

459761 500010

459763 500020

459771 500020

639605 700000

639615 700010

639625 700020

639635 700030

819439 900000

819448 900000

819449 900010

819457 900000

819458 900010

819459 900020

819466 900000

819467 900010

819468 900020

819469 900030

819475 900000

819476 900010

819477 900020

819478 900030

819479 900040

819484 900000

819485 900010

819486 900020

819487 900030

819488 900040

819489 900050

819493 900000

819494 900010

819495 900020

819496 900030

819497 900040

819498 900050

819499 900060

819502 900000

819503 900010

819504 900020

819505 900030

819506 900040

819507 900050

819508 900060

819509 900070

819511 900000

819512 900010

819513 900020

819514 900030

819515 900040

819516 900050

819517 900060

819518 900070

819519 900080

[Me]

(...) <-- admiration

[Lars, a couple of days later – 26^th of December, 2013]

> Hello Eric,

Sort of a "proof" that no sequence can be infinitely long, and the reason that so many sequences terminate with numbers having first digit 4. A similar situation occurs also with first digit 9.

/Lars

--------------------

Let d be the first digit of a number, e the last digit, and k the total number of digits.

Set d=3, k=4, examine all the start numbers 3000...3999 and iterate until d=4.

A pattern emerges:

Start <---- Finish --->

3000

3001 4000 4010

3002 4000 4030

3003 4010

3004 4000 4010

3005

3006 4000 4020

3007 4020

3008 4000 4030

3009 4000 4020

3010

3011 4010

3012 4010

3013 4020

3014 4010

(end of first cycle)

3015

3016 4000 4010

3017 4000 4030

3018 4010

3019 4000 4010

3020

3021 4000 4020

3022 4020

3023 4000 4030

3024 4000 4020

3025

3026 4010

3027 4010

3028 4020

3029 4010

(end of second cycle)

This goes on for a total of 64 cycles (64*15 = 960 values), until we come close to 4000, where the possibility of d=4 starts to appear.

The trailing 40 values are:

3960

3961 4000 4010

3962 4000 4030

3963 4010

3964 4000 4010

3965

3966 4000 4020

3967 4020

3968 4000 4030

3969 4005 4020

3970

3971 4010

3972 4010

3973 4020

3974 4010

3975

3976 4000 4010

3977 4005 4030

3978 4010

3979 4015

3980

3981 4000 4020

3982 4020

3983 4000 4030

3984 4000 4020

3985 4005

3986 4010

3987 4015

3988 4020

3989 4025

3990

3991 4010

3992 4000 4030

3993 4005

3994 4010

3995 4015

3996 4020

3997 4025

3998 4030

3999 4035

In the initial cycles all start numbers (with d=3) lead to numbers with d=4 having e=0 so they terminate.

Among the 40 trailing values, some end in 4005, 4015, 4025 or 4035. These numbers will increment by 20 and eventually end with 5000, 5010 or 5020, so they also terminate.

Examining the start numbers for d=3 and k=5: 30000...39999, the same 15-cycle and the same trailing 40 values appear, although there are now 664 repeats of the cycle. The same is almost certainly true for k>5 too, since the increments between terms is determined by the last digit.

Assume that there exists an infinite sequence. Since the sequence is strictly increasing, and no increment is larger than 81, then terms with d=3, for example, will reappear with successively larger k values.

But we have seen above that all starting values with d=3 will terminate.

Therefore, there can be no infinite sequence, although one may find one that is arbitrarily large.

/Lars

__________

Bravo, Lars! And many thanks!

Have a nice 2014!