Hidden divisors chase

Hello SeqFans,

... let’s start a monotonically increasing S with 1,2:

S = 1,2,...

... if we concatenate 1 and 2 we get ‘12’ which has 1,2,3,4,6,12 as divisors being different. What would be the smallest integer now (to prolong S), which, concatenated with 2, has 5 in its divisors? (This 5 is the "next hidden divisor" we are looking for.)

We try 3 -- but [23] has no 5 among its divisors --,

we try 4 -- but [24] has no 5 among its divisors --,

we try 5 -- and get a hit as [25] is divisible by 5:

S = 1,2,5,...

[12] produces "hidden div." 1,2,3,4,6,12

[25] produces "hidden div." 1,5,25

The next hidden divisor is 7 (we already have 6 thanks to [12]); now, which a(4) would fit S?

This is 6 -- as 56 is divisible by 7:

S = 1,2,5,6,...

We must always keep track of the hidden divisors (HD) we have found so far, in order not to search twice for divisors we already have:

[12] produces HD = 1,2,3,4,6,12

[25] produces HD = 1,5,25

[56] produces HD = 1,2,4,7,8,14,28,56

The next HD we want is thus 9; so a(5) is 12:

S = 1,2,5,6,12,...

[12] produces HD = 1,2,3,4,6,12

[25] produces HD = 1,5,25

[56] produces HD = 1,2,4,7,8,14,28,56

[612] produces HD = 1,2,3,4,6,9,12,17,18,34,36,51,68,102,153,204,306

... etc. I’ve computed (by hand) a few more terms, which I’m sure are wrong at some point (this was late night yesterday – 21st February 2010):

S = 1,2,5,6,12,20,24,31,35,36,54,72,...

Could someone please check and compute a few more terms?

---

BTW:

A similar seq. dealing with HDs would be T:

T = 1,0,2,0,6,...

Where we concatenate _every integer_ printed so far to produce the HDs:

[1] produces HD = 1

[10] produces HD = 1,2,5,10

[102] produces HD = 1,3,6,17,34,102

[1020] produces HD = 1,2,3,4,5,6,10,12,15,17,20,30,34,51,60,68,85,102,170,204,255,340,510

... and to get the next HD (7), I guess we must prolong T with ‘6’...

Lots of fun with HDs!

Best,

É.

__________

The next day Jean-Marc Falcoz was sending me the message hereunder, together with a nice diagram -- thank you, Jean-Marc!

Salut Éric,

Voici 2000 termes de ta suite Hidden divisors of [a(n) and a(n+1) concatenated]. Bravo pour ton calcul à la main, la machine te donne raison jusqu’à 72. En PJ, le graphe des termes avec le graphe superposé de 0.77...*x^2. On voit que ta suite suit (si j’ose dire ;) très bien cette fonction, marrant, non ?

Tout de bon !

J.-M.

S = 1, 2, 5, 6, 12, 20, 24, 31, 35, 36, 54, 72, 90, 117, 145, 151, 164, 200, 203, 218, 221, 235, 267, 270, 300, 328, 384, 410, 454, 461, 472, 500, 550, 566, 618, 695, 753, 760, 796, 800, 822, 913, 934, 1066, 1156, 1235, 1331, 1407, 1420, 1451, 1460, 1550, 1660, 1684, 1775, 1806, 1887, 1890, 2027, 2152, 2156, 2310, 2440, 2510, 2566, 2595, 2697, 2736, 2790, 2872, 2877, 3027, 3030, 3120, 3130, 3181, 3333, 3495, 3668, 3712, 3762, 3768, 3916, 3929, 4154, 4195, 4213, 4330, 4410, 4446, 4650, 4704, 4752, 4946, 5000, 5058, 5157, 5322, 5336, 5565, 5812, 6086, 6300, 6420, 6474, 6682, 6915, 6988, 7239, 7305, 7404, 7512, 7648, 7810, 7932, 7982, 8155, 8252, 8260, 8440, 8641, 8932, 9212, 9334, 9410, 9523, 9657, 9741, 9756, 10144, 10145, 10240, 10434, 10700, 10744, 11080, 11168, 11341, 11384, 11666, 11797, 12163, 12360, 12570, 12860, 13196, 13258, 13357, 13584, 13690, 14006, 14404, 14731, 14785, 14861, 15202, 15550, 15675, 15857, 16290, 16400, 16819, 16836, 17248, 17348, 17682, 17835, 18096, 18462, 18560, 18890, 18950, 19321, 19720, 20132, 20223, 20568, 20821, 20999, 21285, 21728, 21760, 21980, 22090, 22296, 22620, 23024, 23303, 23767, 23832, 24220, 24715, 24779, 25242, 25391, 25864, 26291, 26620, 26654, 27058, 27202, 27395, 27406, 27761, 27770, 27968, 28053, 28202, 28519, 28728, 29254, 29552, 29899, 30336, 30708, 31170, 31296, 31300, 31411, 31535, 31636, 32244, 32636, 32960, 33576, 33620, 34103, 34412, 34885, 35184, 35268, 35566, 36171, 36744, 37133, 37732, 38245, 38406, 39028, 39239, 39331, 39980, 40365, 40888, 41100, 41650, 42245, 42552, 42601, 42775, 42956, 43681, 43780, 44284, 44701, 45062, 45191, 45568, 45613, 46228, 46970, 47680, 47955, 48530, 48877, 49283, 49881, 50426, 51197, 51700, 51956, 52644, 53046, 53616, 54064, 54344, 54850, 55478, 55897, 56351, 56924, 57400, 57957, 57966, 58791, 59196, 59500, 60164, 60595, 61344, 61968, 62768, 63244, 63575, 64396, 65095, 65720, 66572, 66721, 66825, 66912, 67400, 67608, 67793, 68658, 69030, 69628, 69668, 70268, 70752, 71432, 71920, 72790, 73538, 74226, 75018, 75708, 76209, 76395, 76684, 76734, 76752, 77384, 77716, 78488, 78700, 78904, 79262, 79776, 80616, 81280, 82126, 82688, 83224, 83422, 84192, 84420, 84672, 84831, 85746, 86398, 87142, 87257, 87615, 88626, 89280, 89963, 90800, 91232, 91753, 91988, 92312, 93361, 93803, 94621, 94775, 95785, 96652, 97271, 97289, 97762, 97890, 98784, 99390, 99942, 100934, 101664, 101805, 102168, 102865, 103014, 103443, 103746, 103833, 104528, 104908, 105302, 106356, 107052, 107410, 107528, 107598, 108485, 109152, 110265, 110551, 111260, 112002, 112627, 113240, 113344, 113385, 114288, 114690, 115386, 116203, 117176, 118000, 118686, 119508, 119995, 120024, 120516, 120585, 120590, 120894, 121760, 122584, 122984, 123769, 124576, 125232, 125378, 125808, 125970, 126801, 127062, 127808, 128954, 129306, 129774, 130668, 131419, 132532, 132905, 133728, 134410, 134918, 136107, 136737, 136757, 137888, 139136, 139734, 140869, 140930, 141820, 142283, 143200, 143714, 144133, 145241, 145468, 145724, 146047, 146894, 147974, 148970, 149884, 150240, 150330, 150903, 151476, 151794, 151870, 153170, 154139, 155500, 156782, 157455, 158448, 159450, 160418, 160684, 161155, 161222, 162353, 163470, 164520, 165912, 166167, 166467, 167824, 168050, 168393, 168514, 169337, 169558, 170084, 171081, 171338, 172336, 173415, 174400, 175041, 176128, 176239, 176462, 177070, 178448, 178615, 179676, 180331, 180564, 180885, 182198, 183540, 184302, 185580, 186480, 186711, 187337, 187758, 189184, 189912, 191196, 192462, 193425, 193976, 195105, 196080, 196273, 197125, 197603, 198174, 199558, 199758, 200006, 200638, 200814, 202257, 202290, 202782, 202960, 203822, 205035, 205083, 205387, 206167, 207324, 208851, 209714, 210939, 212097, 212272, 212444, 212929, 213497, 213955, 213962, 215560, 216950, 217558, 217935, 218440, 218576, 218861, 220470, 221895, 222584, 223422, 223682, 224166, 225631, 226750, 228324, 229185, 230400, 230486, 230672, 231895, 232016, 232637, 233008, 233560, 234297, 234615, 235680, 237368, 237831, 237834, 238356, 239130, 239425, 240656, 241292, 242068, 242261, 242400, 243330, 244515, 244667, 245180, 245398, 246120, 247840, 248717, 249894, 250080, 251680, 251741, 253312, 253600, 254957, 255163, 255485, 255790, 257345, 258712, 259152, 259650, 260514, 261804, 263575, 264063, 264198, 265383, 267012, 267120, 267573, 269286, 270109, 271970, 273820, 275603, 276137, 277836, 278543, 279430, 280640, 281185, 283050, 283704, 284937, 284984, 285970, 287694, 288596, 289600, 290426, 291322, 292640, 294508, 294764, 296424, 296772, 298388, 299912, 300519, 301587, 302968, 304611, 304905, 304921, 305714, 306440, 306933, 307863, 308071, 309208, 310775, 312057, 313410, 314735, 315500, 316583, 317731, 319430, 320958, 322894, 323830, 324632, 324730, 325416, 325500, 326154, 327918, 328197, 330212, 331771, 333066, 333437, 335231, 335754, 336411, 337332, 338912, 339506, 340390, 340579, 341099, 341208, 342160, 344146, 345257, 346607, 346650, 346657, 348254, 348713, 350307, 352365, 353170, 353376, 353677, 354327, 354744, 354978, 355422, 355540, 356864, 358700, 359570, 360760, 362391, 363742, 365750, 366486, 366576, 367420, 369329, 371460, 372867, 373178, 374746, 375388, 377534, 377875, 380016, 381732, 383902, 385700, 386241, 388095, 390150, 390872, 391572, 391809, 392356, 392499, 392862, 394080, 395244, 396750, 398309, 398848, 399725, 400934, 403012, 403558, 405613, 407492, 408906, 408946, 410819, 412504, 412772, 414856, 416546, 416766, 418712, 420268, 421229, 422790, 423077, 424150, 426284, 426758, 427129, 428744, 429421, 430784, 431686, 433365, 434691, 436288, 437518, 438668, 439378, 440776, 441600, 442894, 443700, 445000, 446000, 446312, 447499, 448000, 448199, 450043, 451550, 451660, 452264, 452593, 454848, 455149, 456960, 457440, 459735, 460044, 461910, 463715, 464988, 467142, 468947, 469918, 470102, 471982, 472615, 475016, 476210, 476341, 477039, 478272, 479951, 481078, 483290, 485677, 487133, 488085, 488726, 491176, 492040, 493403, 494496, 495108, 497000, 497410, 498977, 501442, 503345, 503850, 504248, 506551, 507811, 508216, 510656, 511385, 512485, 513706, 515832, 516164, 516247, 517131, 517276, 519522, 520896, 523392, 525071, 526764, 527181, 528314, 530006, 532093, 533378, 535889, 537703, 538020, 539080, 540600, 543181, 544114, 544855, 546845, 548536, 550028, 550753, 552932, 555280, 556445, 556878, 557925, 559660, 561829, 563093, 563868, 566160, 566750, 569191, 570976, 571748, 573941, 574414, 575941, 575976, 576656, 577075, 578170, 580456, 581879, 583537, 584791, 586325, 588850, 590416, 592927, 594071, 595435, 597880, 600331, 601188, 602070, 602811, 603476, 605392, 607915, 608597, 608831, 609935, 610519, 610777, 612880, 614188, 614521, 615484, 616423, 619160, 619684, 622404, 624122, 625070, 625563, 627115, 627564, 628880, 629975, 632464, 634916, 637032, 639122, 641836, 642898, 644264, 644856, 645447, 646540, 646928, 647038, 649216, 652089, 652107, 654828, 657210, 657831, 659569, 662293, 664403, 666256, 666719, 668890, 669184, 669440, 672160, 674488, 675375, 675986, 676560, 676602, 676812, 677832, 678088, 679072, 681671, 683624, 685324, 686960, 687455, 689936, 692032, 692474, 693637, 694783, 695199, 697116, 698838, 699414, 700010, 700356, 700981, 703150, 704764, 705155, 705829, 707099, 708660, 710882, 713590, 713928, 715280, 716920, 718261, 719160, 719249, 721000, 723352, 725345, 728259, 728336, 729760, 732173, 733540, 736124, 737259, 738685, 739638, 740556, 742860, 743750, 746362, 748738, 751644, 753400, 756156, 756882, 759708, 761093, 761391, 762075, 762688, 763847, 764361, 764732, 767882, 768835, 770840, 773548, 774791, 776120, 776201, 778384, 779666, 780368, 782505, 785055, 788006, 789320, 792039, 795204, 796845, 797838, 797879, 798199, 800777, 802928, 804629, 804997, 806852, 806896, 807894, 810824, 813217, 813464, 814605, 816670, 816776, 818607, 819051, 820160, 822790, 825500, 827008, 828917, 831096, 832296, 832813, 835247, 835620, 836785, 839034, 841829, 844574, 847440, 849035, 851371, 851490, 853653, 856122, 858866, 859382, 860870, 861200, 862984, 865730, 865920, 867735, 868862, 871751, 872593, 873436, 876143, 879154, 880201, 882509, 885100, 886824, 887650, 888986, 889152, 890133, 893402, 895364, 897252, 899510, 902130, 903595, 906525, 908066, 908686, 909454, 910643, 913400, 914349, 914752, 917324, 919750, 922912, 925982, 929430, 930127, 933258, 935322, 935415, 935621, 938282, 938308, 940163, 941600, 944268, 947226, 948018, 949812, 951360, 951556, 953720, 955714, 956678, 957996, 959661, 962900, 963577, 965523, 968590, 969931, 970439, 973174, 975908, 978850, 979466, 982462, 985355, 986125, 988285, 989075, 990090, 991908, 993702, 996372, 999133, 1000212, 1002748, 1005500, 1005885, 1008669, 1011400, 1012857, 1013376, 1014936, 1017159, 1018371, 1018983, 1021280, 1021712, 1024429, 1025950, 1029175, 1030469, 1032844, 1034387, 1036080, 1037160, 1037798, 1040248, 1042554, 1043166, 1046466, 1046931, 1048848, 1050860, 1051782, 1054230, 1057518, 1058744, 1061064, 1061428, 1063680, 1066957, 1068405, 1068842, 1072216, 1075091, 1075660, 1077041, 1079580, 1079945, 1081617, 1082233, 1083824, 1084825, 1087473, 1087535, 1091056, 1092384, 1095000, 1097301, 1099001, 1099338, 1100244, 1103342, 1105936, 1106623, 1109900, 1112128, 1115144, 1117180, 1118588, 1120064, 1121314, 1123558, 1126550, 1130236, 1131473, 1131977, 1132418, 1133496, 1134288, 1137834, 1138686, 1139420, 1140318, 1142804, 1143712, 1147020, 1150103, 1152987, 1153245, 1154283, 1158144, 1160692, 1161114, 1162880, 1165124, 1167310, 1170456, 1174362, 1175501, 1179146, 1180720, 1182286, 1182399, 1184997, 1188573, 1188816, 1189337, 1191496, 1195509, 1195866, 1199436, 1202313, 1202771, 1204900, 1206759, 1207365, 1208052, 1211985, 1213680, 1215251, 1217575, 1218064, 1220510, 1223572, 1225928, 1227039, 1228076, 1229441, 1230340, 1234007, 1234458, 1236104, 1237886, 1240410, 1243031, 1246874, 1249393, 1249878, 1253638, 1257724, 1261556, 1264640, 1265602, 1265855, 1269760, 1272816, 1276073, 1279172, 1282864, 1284580, 1284831, 1286138, 1286675, 1289980, 1293172, 1293696, 1297733, 1298685, 1300896, 1303125, 1304100, 1306898, 1307360, 1308190, 1312026, 1312066, 1312619, 1316115, 1319556, 1321445, 1322820, 1324960, 1326162, 1328748, 1329444, 1330240, 1334148, 1338248, 1341059, 1342288, 1342514, 1345770, 1348806, 1352932, 1353732, 1355019, 1357816, 1358168, 1358451, 1361311, 1361700, 1363572, 1365596, 1367929, 1370162, 1371460, 1374656, 1375117, 1376980, 1380342, 1383318, 1387022, 1388340, 1391643, 1395620, 1397187, 1401000, 1402683, 1406364, 1409597, 1410170, 1412634, 1416425, 1419406, 1420820, 1422812, 1426789, 1428467, 1431436, 1434164, 1437095, 1438336, 1439131, 1442980, 1444019, 1444689, 1445628, 1447371, 1451046, 1452140, 1452922, 1454368, 1454869, 1457278, 1457580, 1457962, 1461392, 1464857, 1467538, 1469235, 1470920, 1474816, 1475480, 1478545, 1478914, 1482005, 1483125, 1485126, 1485522, 1486560, 1489803, 1492552, 1493636, 1495858, 1496558, 1500657, 1502650, 1505092, 1507006, 1510936, 1512911, 1513285, 1515257, 1515652, 1518673, 1521871, 1522000, 1523292, 1527390, 1527822, 1529310, 1533402, 1536840, 1538329, 1542813, 1545270, 1548655, 1550639, 1553353, 1555663, 1556024, 1557680, 1560376, 1560568, 1562246, 1565399, 1567488, 1570683, 1574958, 1576786, 1577552, 1580530, 1581167, 1582135, 1586630, 1589514, 1593735, 1595832, 1597900, 1599476, 1600378, 1604864, 1605711, 1606700, 1606876, 1611304, 1612946, 1615845, 1620376, 1623910, 1624145, 1628167, 1630782, 1634172, 1638268, 1638852, 1643064, 1646112, 1646424, 1649273, 1650052, 1653543, 1657014, 1659794, 1664460, 1668494, 1669014, 1669725, 1672740, 1673055, 1673153, 1675234, 1679303, 1681447, 1684048, 1686421, 1687695, 1688861, 1692552, 1692640, 1692809, 1693158, 1696887, 1697150, 1701679, 1702014, 1702780, 1704580, 1705694, 1707387, 1710302, 1710331, 1713464, 1717921, 1720844, 1724438, 1727350, 1727856, 1731594, 1731916, 1732327, 1736990, 1738266, 1740265, 1743480, 1744180, 1748784, 1752664, 1755680, 1759168, 1763567, 1766995, 1767629, 1772050, 1775488, 1776125, 1777970, 1781156, 1781689, 1783655, 1785148, 1785225, 1786962, 1787633, 1789232, 1790207, 1791855, 1792872, 1793994, 1796452, 1797075, 1797324, 1798081, 1800989, 1803386, 1805507, 1806882, 1810780, 1812548, 1813046, 1817712, 1821180, 1825221, 1827696, 1831830, 1832873, 1835665, 1840520, 1841305, 1844528, 1848363, 1851976, 1855040, 1856290, 1857841, 1858710, 1860845, 1863692, 1865328, 1866667, 1868810, 1873000, 1877369, 1878540, 1879719, 1881914, 1882441, 1882680, 1887147, 1889850, 1894767, 1897620, 1898990, 1900564, 1902950, 1903825, 1908443, 1909240, 1910984, 1915736, 1918638, 1923009, 1926436, 1928560, 1930644, 1935451, 1937168, 1941270, 1941800, 1945075, 1948329, 1952553, 1954905, 1959478, 1964100, 1968036, 1968502, 1972190, 1976939, 1977235, 1980532, 1983340, 1986853, 1991513, 1994344, 1994624, 1998632, 1998695, 1998739, 2003240, 2004726, 2005774, 2008690, 2011008, 2011208, 2015240, 2015872, 2018309, 2021584, 2025530, 2030172, 2030330, 2031726, 2035920, 2040292, 2044995, 2047714, 2047852, 2049321, 2054032, 2056430, 2058282, 2062874, 2062921, 2064230, 2067392, 2069256, 2073332, 2073970, 2078492, 2079557, 2080699, 2085324, 2086655, 2090352, 2094445, 2097244, 2101180, 2101290, 2102184, 2104918, 2109803, 2109888, 2113953, 2118920, 2122441, 2125431, 2128774, 2133940, 2136383, 2140023, 2145061, 2149373, 2153890, 2154084, 2158847, 2160865, 2161330, 2161742, 2166081, 2167080, 2171378, 2174105, 2177610, 2181108, 2181712, 2182586, 2186930, 2187901, 2192283, 2196402, 2196765, 2199570, 2204076, 2204648, 2209115, 2212796, 2215256, 2218734, 2220636, 2222889, 2224686, 2225216, 2226208, 2226393, 2231271, 2233242, 2234374, 2236112, 2237802, 2241566, 2243403, 2248220, 2249728, 2251298, 2255257, 2260504, 2265722, 2269315, 2272950, 2276450, 2280106, 2283718, 2286644, 2290530, 2295136, 2296111, 2299966, 2305104, 2309535, 2314496, 2315195, 2316161, 2317742, 2323131, 2327685, 2329848, 2333865, 2335700, 2340000, 2340226, 2340350, 2344876, 2345925, 2347866, 2350503, 2353186, 2354983, 2356978, 2358772, 2360242, 2365090, 2369367, 2373108, 2378558, 2382031, 2387345, 2389002, 2392730, 2396403, 2397982, 2399258, 2403982, 2405599, 2410870, 2411198, 2413632, 2414820, 2417027, 2421591, 2425060, 2429366, 2432941, 2437056, 2437890, 2439384, 2440704, 2444140, 2444260, 2448700, 2453399, 2453740, 2455070, 2455240, 2455328, 2459328, 2462516, 2465865, 2470760, 2472211, 2475825, 2478484, 2481272, 2483760, 2483772, 2488470, 2494020, 2497630, 2500664, 2502048, 2507606, 2508640, 2511100, 2511753, 2517168, 2520785, 2525222, 2526547, 2529442, 2532382, 2534802, 2537574, 2538945, 2543608, 2547947, 2550473, 2555872, 2557550, 2558516, 2559680, 2565112, 2565695, 2567057, 2567834, 2572937, 2574928, 2576576, 2582292, 2586592, 2586854, 2587872, 2593493, 2595556, 2599375, 2602358, 2606518, 2607608, 2612196, 2612499, 2618298, 2621530, 2626694, 2630120, 2632874, 2633437, 2634804, 2638425, 2639756, 2640020, 2642422, 2644586, 2650032, 2651250, 2651580, 2654942, 2657413, 2662552, 2667682, 2669728, 2673962, 2675390, 2679195, 2684112, 2684321, 2686136, 2688913, 2689257, 2690312, 2691164, 2691770, 2694233, 2696824, 2698796, 2703744, 2708987, 2714503, 2716235, 2717773, 2718800, 2723455, 2727752, 2732988, 2738742, 2744654, 2745514, 2746694, 2750547, 2751216, 2756883, 2758870, 2762630, 2768168, 2773500, 2774489, 2779432, 2782898, 2786044, 2786550, 2790235, 2791538, 2796266, 2801360, 2802073, 2803248, 2807120, 2808436, 2813216, 2814600, 2817025, 2821424, 2821830, 2822493, 2825440, 2831286, 2834705, 2836510, 2840607, 2845005, 2849000, 2854264, 2860088, 2860725, 2864400, 2864874, 2865952, 2869560, 2869714, 2872703, 2877808, 2878183, 2880612, 2883827, 2888665, 2893927, 2895233, 2898720, 2900526, 2900597, 2905772, 2908494, 2912025, 2915564, 2915896, 2916056, 2916804, 2922536, 2925308, 2930380, 2936216, 2941403, 2943361, 2945560, 2948200, 2950228, 2950252, 2951537, 2954950, 2959370, 2962694, 2967028, 2970716, 2976517, 2977792, 2979434, 2981436, 2982135, 2987345, 2989166, 2992468, 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« Le graphe des termes, superposé au graphe de [0.77...*x^2] »

Merci encore, Jean-Marc !

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