Divisor Runs

 My diurnal age today, 26885, is a member of OEIS A006601:

A006601

Numbers \(k\) such that \(k\), \(k+1\), \(k+2\) and \(k+3\) have the same number of divisors.

This means that 26885, 26886, 26887, 26888 and 26889 have the same number of divisors. Let's check that this is the case:

26885 = 5 x 19 x 283 and has 8 divisors

26886 = 2 x 3 x 4481 and has 8 divisors

26887 = 7 x 23 x 167 and has 8 divisors

26888 = 2^3 x 3361 and has 8 divisors

Let's not forget the rule for determining the number of divisors from the factorisation: add one to the index of each prime factor and then multiply them together. Runs of four numbers with the same number of divisors are rare. 

Below are listed the numbers up to one million, all members of the OEIS sequence (permalink). There are only 1171 of them, representing 0.1171% of the numbers in the range. 

242, 3655, 4503, 5943, 6853, 7256, 8392, 9367, 10983, 11605, 11606, 12565, 12855, 12856, 12872, 13255, 13782, 13783, 14312, 16133, 17095, 18469, 19045, 19142, 19143, 19940, 20165, 20965, 21368, 21494, 21495, 21512, 22855, 23989, 26885, 28135, 28374, 28375, 28376, 29605, 30583, 31735, 31910, 32005, 32792, 33062, 33608, 33845, 34069, 36392, 37256, 40311, 40312, 41335, 42805, 42806, 43304, 43526, 43766, 44213, 45686, 45733, 47845, 48054, 49147, 49765, 50582, 50583, 51752, 54103, 54585, 54966, 55063, 55254, 55255, 55976, 56343, 58952, 59815, 60231, 60232, 60663, 60664, 61142, 62343, 65334, 66952, 67015, 68104, 69303, 71095, 73927, 74053, 76262, 76982, 77432, 78535, 78872, 79094, 79095, 79591, 80726, 82855, 84469, 86887, 87655, 87656, 87896, 90181, 90182, 90183, 91495, 93063, 94262, 94645, 95384, 95414, 95512, 95845, 95846, 97255, 98102, 98984, 99655, 99656, 100711, 100952, 101125, 103352, 103621, 103622, 104222, 104744, 104870, 105301, 107365, 108902, 109191, 109765, 109766, 112567, 113942, 115591, 115592, 115912, 116965, 117032, 118069, 118261, 118615, 118923, 120727, 120728, 120965, 121045, 121046, 122151, 122152, 122871, 122872, 123944, 124663, 125335, 129829, 130135, 131815, 133624, 134582, 136375, 136825, 139863, 141654, 142454, 142455, 142806, 142807, 143365, 145352, 146936, 151285, 152102, 152552, 152630, 152631, 153461, 153543, 153703, 153992, 157493, 157494, 157495, 157910, 158216, 158342, 160934, 162295, 164982, 165542, 166791, 167671, 169112, 169141, 169813, 171893, 171894, 171895, 171896, 172501, 173893, 173912, 174054, 174055, 174872, 175143, 175144, 178086, 178087, 180901, 180902, 180965, 180966, 180967, 180968, 181207, 182215, 183205, 183206, 183554, 183685, 184327, 184328, 185815, 186231, 186951, 186952, 188293, 188294, 191943, 191944, 192728, 194695, 197463, 198776, 198806, 199111, 199112, 199255, 199733, 199832, 201512, 201943, 203432, 203621, 204323, 204324, 204806, 206390, 206552, 210133, 210134, 210135, 211592, 212005, 212006, 214885, 217144, 217526, 218695, 219063, 220232, 220405, 223687, 223861, 223976, 224005, 224776, 225463, 225464, 226792, 227576, 229015, 229352, 229685, 230167, 230869, 231413, 233942, 234615, 235112, 235495, 235496, 236006, 236245, 236246, 236792, 237031, 237063, 238694, 240904, 243415, 243445, 244742, 246294, 246631, 246632, 248311, 248504, 249445, 249512, 251751, 251752, 252967, 253014, 253015, 258008, 259205, 261734, 261992, 262661, 262932, 263765, 263766, 264296, 264952, 266485, 266869, 267655, 268021, 268933, 269912, 270181, 270613, 271526, 272455, 273368, 274088, 277543, 277624, 279302, 279542, 280230, 280712, 282055, 282245, 282584, 283496, 284581, 285413, 287029, 287704, 287814, 288085, 290981, 291781, 292853, 293029, 294085, 295014, 295015, 295352, 296453, 298694, 298695, 298696, 298903, 299863, 301062, 301189, 301622, 302103, 304262, 304675, 305335, 306086, 306631, 306632, 306805, 307254, 307255, 307285, 307445, 308342, 309415, 309416, 309655, 309656, 309830, 310503, 310808, 312151, 312152, 316326, 319765, 321304, 321543, 322214, 324662, 326791, 326792, 328262, 328263, 328933, 329125, 329144, 329432, 331285, 331638, 332102, 332103, 332888, 333176, 334085, 335093, 335863, 336485, 337254, 337335, 337431, 341030, 341031, 341463, 341605, 343189, 343623, 344935, 345782, 346502, 346503, 346504, 350312, 352135, 352983, 353605, 353606, 354056, 355453, 356293, 356504, 358015, 358309, 358934, 360055, 360183, 361275, 361285, 361322, 362341, 362534, 363207, 365269, 365384, 366005, 366535, 369061, 369062, 369063, 369127, 370165, 372711, 373143, 373784, 374630, 374888, 376453, 376742, 376743, 376744, 377095, 377896, 379255, 379670, 379862, 380344, 381592, 381702, 382854, 383872, 384391, 386726, 386965, 386966, 387302, 387703, 390470, 390533, 393589, 393653, 394454, 395462, 395704, 396902, 397352, 398408, 399368, 399655, 399656, 400165, 400166, 400375, 400693, 400694, 401270, 401653, 402294, 402295, 403526, 404391, 404392, 406135, 406136, 407605, 411062, 412615, 413176, 414421, 417445, 417446, 417608, 418135, 420005, 420006, 422885, 422886, 423031, 424712, 425576, 426584, 427015, 427352, 427494, 428053, 428392, 430645, 431509, 431895, 432085, 432086, 432392, 433064, 434552, 434584, 435016, 437671, 439016, 440869, 445062, 446295, 446296, 447365, 447366, 448903, 451333, 451733, 451784, 451910, 452215, 453895, 455365, 456952, 459734, 460742, 460805, 461511, 461512, 461671, 464214, 464582, 464583, 464871, 464872, 465031, 465032, 465544, 465589, 467030, 467031, 468902, 469864, 471367, 472070, 472853, 475736, 475976, 476377, 479845, 479846, 480134, 480664, 481061, 482693, 482744, 483941, 486229, 486487, 486962, 487191, 487192, 490375, 490855, 493382, 493383, 494744, 494934, 495590, 496792, 500312, 501845, 501846, 502183, 506245, 506821, 506905, 507125, 507782, 507783, 507944, 508712, 515192, 517304, 517431, 517432, 519366, 519590, 520087, 520088, 520807, 521126, 521461, 525064, 525845, 525846, 528853, 528902, 530168, 530870, 530887, 531445, 531446, 532741, 532742, 532983, 533703, 533767, 535064, 536167, 536583, 538374, 538453, 539270, 539911, 539912, 541045, 542792, 543062, 543063, 545912, 547192, 548536, 548776, 549590, 549992, 551192, 552390, 554741, 554742, 555032, 555302, 556743, 556869, 558086, 558229, 558230, 559013, 559445, 559688, 560791, 560792, 562328, 564008, 564053, 565014, 567302, 569864, 570053, 570533, 571621, 571863, 572214, 572215, 572405, 572744, 573365, 574743, 574885, 574886, 576895, 579062, 580935, 580982, 581509, 583286, 583383, 585494, 586885, 586952, 587462, 587864, 588485, 588728, 589208, 589592, 589765, 590935, 592376, 594344, 596071, 596245, 596533, 597736, 598087, 600294, 604262, 605432, 606343, 607526, 610310, 610311, 610312, 610904, 611384, 613192, 614965, 615061, 615352, 616063, 616135, 616375, 617096, 617125, 617896, 619722, 619832, 621445, 624054, 627654, 627735, 628664, 628855, 628856, 631432, 631832, 632694, 632695, 633110, 634232, 634262, 636710, 638407, 638408, 639734, 639991, 639992, 640855, 640885, 640886, 641765, 642295, 644293, 645031, 645205, 646232, 646789, 647381, 647382, 647383, 647384, 648326, 648344, 649623, 649624, 649830, 649856, 650245, 650741, 652885, 654182, 654952, 656312, 656934, 657542, 657895, 658232, 658711, 659701, 662390, 662485, 664183, 664184, 665093, 665816, 668211, 668632, 669736, 672536, 676165, 676262, 676711, 678486, 679735, 679736, 680726, 680869, 682070, 684805, 685864, 688134, 689143, 689432, 690805, 693031, 693032, 693542, 693685, 694504, 694741, 694742, 697206, 700134, 700135, 701463, 701544, 701911, 701912, 702182, 702183, 703335, 705416, 706069, 707032, 707286, 707287, 707288, 709303, 710454, 711752, 712405, 712453, 713845, 714632, 716005, 716006, 718232, 718471, 720085, 720245, 720246, 722055, 723991, 723992, 725845, 726565, 726566, 727094, 729542, 729543, 729544, 730645, 731461, 731462, 731703, 731941, 733158, 734696, 734948, 736070, 738902, 740310, 741254, 741783, 742552, 742567, 742855, 744005, 744709, 744806, 751142, 751303, 754743, 755191, 755192, 755462, 756343, 756581, 757207, 757544, 757670, 760504, 760711, 760712, 760855, 762086, 762296, 763688, 763734, 763735, 763862, 765032, 765894, 766263, 766335, 766982, 767431, 767893, 768776, 769189, 769862, 769863, 769864, 770693, 771445, 771446, 772213, 772214, 772711, 776936, 777254, 777271, 777415, 778294, 778981, 780854, 781832, 782312, 782742, 783416, 783894, 783895, 786565, 786566, 787735, 787736, 788504, 788583, 789832, 789895, 792326, 793192, 796405, 796502, 796503, 798470, 799255, 802086, 802855, 802856, 802904, 803605, 805862, 807079, 807445, 809301, 809302, 809462, 810902, 810903, 812215, 812821, 813942, 813943, 817592, 820262, 820263, 820310, 820471, 820856, 822151, 823285, 823432, 823862, 824629, 824744, 825735, 828872, 829445, 829622, 832502, 832503, 833192, 836389, 836390, 836407, 836632, 837543, 838405, 838615, 839030, 839991, 839992, 841143, 841592, 842005, 842006, 842101, 842373, 844133, 844231, 844232, 845221, 845382, 845383, 845815, 846744, 846902, 846966, 846967, 847254, 847255, 848246, 848582, 848583, 848695, 848966, 849493, 849494, 850088, 852296, 853095, 853096, 854018, 855320, 855703, 857606, 857814, 857815, 858470, 862616, 863528, 864184, 865013, 865526, 866341, 866342, 868567, 868614, 869096, 871381, 871382, 875095, 875462, 876152, 876295, 876296, 876631, 877928, 880119, 880662, 882183, 883351, 883352, 883832, 884504, 884821, 885512, 887576, 889189, 889816, 890534, 891271, 891415, 891776, 893126, 893512, 894952, 896485, 898328, 898374, 898645, 898885, 901503, 903351, 903512, 904405, 904712, 905511, 905512, 905815, 906967, 907464, 908631, 908632, 909176, 911270, 911912, 915781, 915896, 916373, 917527, 919207, 923989, 927624, 928232, 930181, 930390, 934311, 934312, 937025, 938203, 939416, 939894, 940312, 941432, 942485, 943285, 943815, 944407, 944485, 945783, 946712, 947703, 947767, 948965, 950005, 950342, 950343, 951205, 952039, 952552, 954392, 956552, 957414, 957512, 958645, 958791, 958792, 959365, 959846, 960967, 960968, 961301, 961494, 962965, 963445, 964310, 969832, 970616, 970808, 980184, 981205, 984135, 984295, 984344, 986455, 987032, 988374, 988375, 988952, 989095, 989894, 990661, 990662, 991045, 991622, 991910, 992693, 992870, 993542, 995815, 995942, 996229, 998389, 999649

Naturally I was interested in finding the limit of these runs from one to one million. What about runs of five numbers? Well the number shrinks drastically to just 179 (permalink).

11605, 12855, 13782, 19142, 21494, 28374, 28375, 40311, 42805, 50582, 55254, 60231, 60663, 79094, 87655, 90181, 90182, 95845, 99655, 103621, 109765, 115591, 120727, 121045, 122151, 122871, 142454, 142806, 152630, 157493, 157494, 171893, 171894, 171895, 174054, 175143, 178086, 180901, 180965, 180966, 180967, 183205, 184327, 186951, 188293, 191943, 199111, 204323, 210133, 210134, 212005, 225463, 235495, 236245, 246631, 251751, 253014, 263765, 295014, 298694, 298695, 306631, 307254, 309415, 309655, 312151, 326791, 328262, 332102, 341030, 346502, 346503, 353605, 369061, 369062, 376742, 376743, 386965, 399655, 400165, 400693, 402294, 404391, 406135, 417445, 420005, 422885, 432085, 446295, 447365, 461511, 464582, 464871, 465031, 467030, 479845, 487191, 493382, 501845, 507782, 517431, 520087, 525845, 531445, 532741, 539911, 543062, 554741, 558229, 560791, 572214, 574885, 610310, 610311, 628855, 632694, 638407, 639991, 640885, 647381, 647382, 647383, 649623, 664183, 679735, 693031, 694741, 700134, 701911, 702182, 707286, 707287, 716005, 720245, 723991, 726565, 729542, 729543, 731461, 755191, 760711, 763734, 769862, 769863, 771445, 772213, 783894, 786565, 787735, 796502, 802855, 809301, 810902, 813942, 820262, 832502, 836389, 839991, 842005, 844231, 845382, 846966, 847254, 848582, 849493, 853095, 857814, 866341, 871381, 876295, 883351, 905511, 908631, 934311, 950342, 958791, 960967, 988374, 990661

What about runs of six numbers? There are just 18 (permalink).

28374, 90181, 157493, 171893, 171894, 180965, 180966, 210133, 298694, 346502, 369061, 376742, 610310, 647381, 647382, 707286, 729542, 769862

What about runs of seven numbers? There are a mere three (permalink) and there are no runs of eight numbers in the range up to one million.

171893, 180965, 647381

Let's look at the factorisation of these numbers. We find all three have eight divisors and most are sphenic numbers except for a single number that is divisible by eight.

171893 = 19 x 83 x 109 and has 8 divisors

171894 = 2 x 3 x 28649 and has 8 divisors

171895 = 5 x 31 x 1109 and has 8 divisors

171896 = 2^3 x 21487 and has 8 divisors

171897 = 3 x 11 x 5209 and has 8 divisors

171898 = 2 x 61 x 1409 and has 8 divisors

171899 = 7 x 13 x 1889 and has 8 divisors

180965 = 5 x 17 x 2129 and has 8 divisors

180966 = 2 x 3 x 30161 and has 8 divisors

180967 = 37 x 67 x 73 and has 8 divisors

180968 = 2^3 x 22621 and has 8 divisors

180969 = 3 x 179 x 337 and has 8 divisors

180970 = 2 x 5 x 18097 and has 8 divisors

180971 = 7 x 103 x 251 and has 8 divisors

647381 = 7 x 23 x 4021 and has 8 divisors

647382 = 2 x 3 x 107897 and has 8 divisors

647383 = 11 x 229 x 257 and has 8 divisors

647384 = 2^3 x 80923 and has 8 divisors

647385 = 3 x 5 x 43159 and has 8 divisors

647386 = 2 x 89 x 3637 and has 8 divisors

647387 = 13 x 19 x 2621 and has 8 divisors

The appearance of a number with eight as a divisor is not surprising given that, out of eight consecutive numbers, one must be divisible by eight. In fact every fourth number must be divisible by 4 and it appears that in all our runs the number immediately before the first number and after the last number is divisible by 4. For example, consider the case of 647381:

647380 = 2^2 * 5 * 32369 and has 12 divisors

647381 = 7 * 23 * 4021 and has 8 divisors

647382 = 2 * 3 * 107897 and has 8 divisors

647383 = 11 * 229 * 257 and has 8 divisors

647384 = 2^3 * 80923 and has 8 divisors

647385 = 3 * 5 * 43159 and has 8 divisors

647386 = 2 * 89 * 3637 and has 8 divisors

647387 = 13 * 19 * 2621 and has 8 divisors

647388 = 2^2 * 3^2 * 7^2 * 367 and has 54 divisors

By extension, one can investigate other sorts of runs. For example, runs of numbers with the same number of prime factors.  Investigation reveals that 526095 is the only number in the range up to one million that starts off a run of 14 numbers, all with three prime factors (permalink).

526095 = 3^5 x 5 x 433 has prime factors [3, 5, 433]

526096 = 2^4 x 131 x 251 has prime factors [2, 131, 251]

526097 = 11 x 13^2 x 283 has prime factors [11, 13, 283]

526098 = 2 x 3 x 87683 has prime factors [2, 3, 87683]

526099 = 7 x 17 x 4421 has prime factors [7, 17, 4421]

526100 = 2^2 x 5^2 x 5261 has prime factors [2, 5, 5261]

526101 = 3 x 31 x 5657 has prime factors [3, 31, 5657]

526102 = 2 x 23 x 11437 has prime factors [2, 23, 11437]

526103 = 37 x 59 x 241 has prime factors [37, 59, 241]

526104 = 2^3 x 3^2 x 7307 has prime factors [2, 3, 7307]

526105 = 5 x 43 x 2447 has prime factors [5, 43, 2447]

526106 = 2 x 7 x 37579 has prime factors [2, 7, 37579]

526107 = 3 x 157 x 1117 has prime factors [3, 157, 1117]

526108 = 2^2 x 11^2 x 1087 has prime factors [2, 11, 1087]