Iban Numbers

I was struggling to find something of significance (in my mind) about the number associated with my diurnal age today which is 27147. However, after much fruitless investigation and experimentation, I noticed something at the very bottom of the Numbers Aplenty entry for the number. It read as follows:

The spelling of 27147 in words is "twenty-seven thousand, one hundred forty-seven", and thus it is an iban number.

Hmmm. What on Earth is an iban number I thought. Well a definition was only a hyperlink away:

A number is called iban if its name (in English) does not contain the letter "i". Assuming that the name of every power of 10 greater than  $10^5$  ends in "-illion" (like million, billion, trillion, etc.), then the iban numbers are finite. Counting 0 (zero) there are 30276 of them, the largest being 777777. Iban numbers belong to the same family as aban numbers, eban numbers, oban numbers, and uban numbers. 

These numbers constitute OEIS A089589:

A089589

Iban numbers (the letter i is banned from the English name of the number).

The initial members are:

0, 1, 2, 3, 4, 7, 10, 11, 12, 14, 17, 20, 21, 22, 23, 24, 27, 40, 41, 42, 43, 44, 47, 70, 71, 72, 73, 74, 77, 100, 101, 102, 103, 104, 107, 110, 111, 112, 114, 117, 120, 121, 122, 123, 124, 127, 140, 141, 142, 143, 144, 147, 170, 171, 172, 173, 174, 177, 200, 201

The OEIS comments include the following Python code:

from itertools import islice

from num2words import num2words

def agen(): yield from (k for k in range(10**6) if "i" not in num2words(k))

print(list(islice(agen(), 60)))

This doesn't work so I asked Google's Bard to fix the problem and it said to add the line "import num2words". This gives the following code: 

import num2words
from itertools import islice
from num2words import num2words
def agen(): yield from (k for k in range(10**6) if "i" not in num2words(k))
print(list(islice(agen(), 60)))

This code actually works using SageMath on my laptop and generates the entire 30276 numbers by replacing the 60. However, it still won't run on SageMathCell or online Python compiler like Programitz.

The num2words works as shown in Figure 1:

Figure 1

 There are many iban numbers in the range between 27000 and 28000. Here they are:

27000, 27001, 27002, 27003, 27004, 27007, 27010, 27011, 27012, 27014, 27017, 27020, 27021, 27022, 27023, 27024, 27027, 27040, 27041, 27042, 27043, 27044, 27047, 27070, 27071, 27072, 27073, 27074, 27077, 27100, 27101, 27102, 27103, 27104, 27107, 27110, 27111, 27112, 27114, 27117, 27120, 27121, 27122, 27123, 27124, 27127, 27140, 27141, 27142, 27143, 27144, 27147, 27170, 27171, 27172, 27173, 27174, 27177, 27200, 27201, 27202, 27203, 27204, 27207, 27210, 27211, 27212, 27214, 27217, 27220, 27221, 27222, 27223, 27224, 27227, 27240, 27241, 27242, 27243, 27244, 27247, 27270, 27271, 27272, 27273, 27274, 27277, 27300, 27301, 27302, 27303, 27304, 27307, 27310, 27311, 27312, 27314, 27317, 27320, 27321, 27322, 27323, 27324, 27327, 27340, 27341, 27342, 27343, 27344, 27347, 27370, 27371, 27372, 27373, 27374, 27377, 27400, 27401, 27402, 27403, 27404, 27407, 27410, 27411, 27412, 27414, 27417, 27420, 27421, 27422, 27423, 27424, 27427, 27440, 27441, 27442, 27443, 27444, 27447, 27470, 27471, 27472, 27473, 27474, 27477, 27700, 27701, 27702, 27703, 27704, 27707, 27710, 27711, 27712, 27714, 27717, 27720, 27721, 27722, 27723, 27724, 27727, 27740, 27741, 27742, 27743, 27744, 27747, 27770, 27771, 27772, 27773, 27774, 27777

Prior to 27000, the last iban number is 24777 and after 27777, the next is 40000.  Figure 2 shows a plot of the iban numbers.

Figure 2

While we're at it, we may as well look at similar types of numbers. Let's start with aban numbers. Numbers Aplenty defines these as follows:

A number is called aban if its name (in English) does not contain the letter "a". The word "and" is not counted and in general I do not use it when I spell out numbers. Among the words used to construct numbers names, only the word "thousand" contains an "a" so the aban numbers are the numbers from 1 to 999, from 1000000 to 1000999, from 2000000 to 2000999, and so on. The sum of the reciprocals of aban numbers does not converge and grows slowlytowards infinity.

Figure 3 shows a graph of the initial aban numbers up to 1000 which are:

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 200, 300, 400, 500, 600, 700, 800, 900

Figure 2

Next we'll consider eban numbers defined as follows by Numbers Aplenty:

A number is called eban if its name (in English) does not contain the letter "e".

It is easy to see that the eban numbers are all even and their last two digits must be one of 02, 04, 06, 30, 32, 34, 36, 40, 42, 44, 46, 50, 52, 54, 56, 60, 62, 64, or 66.

Here are the initial members and Figure 3 shows a plot of these numbers:

2, 4, 6, 30, 32, 34, 36, 40, 42, 44, 46, 50, 52, 54, 56, 60, 62, 64, 66, 2000, 2002, 2004, 2006, 2030, 2032, 2034, 2036, 2040, 2042, 2044, 2046, 2050, 2052, 2054, 2056, 2060, 2062, 2064, 2066, 4000, 4002, 4004, 4006, 4030, 4032, 4034, 4036, 4040, 4042, 4044, 4046, 4050, 4052, 4054, 4056, 4060, 4062, 4064, 4066, 6000, 6002, 6004, 6006, 6030, 6032, 6034, 6036, 6040, 6042, 6044, 6046, 6050, 6052, 6054, 6056, 6060, 6062, 6064, 6066, 30000, 30002, 30004, 30006, 30030, 30032, 30034, 30036, 30040, 30042, 30044, 30046, 30050, 30052, 30054, 30056, 30060, 30062, 30064, 30066, 32000, 32002, 32004, 32006, 32030, 32032, 32034, 32036, 32040, 32042, 32044, 32046, 32050, 32052, 32054, 32056, 32060, 32062, 32064, 32066, 34000, 34002, 34004, 34006, 34030, 34032, 34034, 34036, 34040, 34042, 34044, 34046, 34050, 34052, 34054, 34056, 34060, 34062, 34064, 34066, 36000, 36002, 36004, 36006, 36030, 36032, 36034, 36036, 36040, 36042, 36044, 36046, 36050, 36052, 36054, 36056, 36060, 36062, 36064, 36066, 40000, 40002, 40004, 40006, 40030, 40032, 40034, 40036, 40040, 40042, 40044, 40046, 40050, 40052, 40054, 40056, 40060, 40062, 40064, 40066, 42000, 42002, 42004, 42006, 42030, 42032, 42034, 42036, 42040, 42042, 42044, 42046, 42050, 42052, 42054, 42056, 42060, 42062, 42064, 42066, 44000, 44002, 44004, 44006, 44030, 44032, 44034, 44036, 44040, 44042, 44044, 44046, 44050, 44052, 44054, 44056, 44060, 44062, 44064, 44066, 46000, 46002, 46004, 46006, 46030, 46032, 46034, 46036, 46040, 46042, 46044, 46046, 46050, 46052, 46054, 46056, 46060, 46062, 46064, 46066, 50000, 50002, 50004, 50006, 50030, 50032, 50034, 50036, 50040, 50042, 50044, 50046, 50050, 50052, 50054, 50056, 50060, 50062, 50064, 50066, 52000, 52002, 52004, 52006, 52030, 52032, 52034, 52036, 52040, 52042, 52044, 52046, 52050, 52052, 52054, 52056, 52060, 52062, 52064, 52066, 54000, 54002, 54004, 54006, 54030, 54032, 54034, 54036, 54040, 54042, 54044, 54046, 54050, 54052, 54054, 54056, 54060, 54062, 54064, 54066, 56000, 56002, 56004, 56006, 56030, 56032, 56034, 56036, 56040, 56042, 56044, 56046, 56050, 56052, 56054, 56056, 56060, 56062, 56064, 56066, 60000, 60002, 60004, 60006, 60030, 60032, 60034, 60036, 60040, 60042, 60044, 60046, 60050, 60052, 60054, 60056, 60060, 60062, 60064, 60066, 62000, 62002, 62004, 62006, 62030, 62032, 62034, 62036, 62040, 62042, 62044, 62046, 62050, 62052, 62054, 62056, 62060, 62062, 62064, 62066, 64000, 64002, 64004, 64006, 64030, 64032, 64034, 64036, 64040, 64042, 64044, 64046, 64050, 64052, 64054, 64056, 64060, 64062, 64064, 64066, 66000, 66002, 66004, 66006, 66030, 66032, 66034, 66036, 66040, 66042, 66044, 66046, 66050, 66052, 66054, 66056, 66060, 66062, 66064, 66066

Figure 3

This leads on to the oban numbers defined as follows by Numbers Aplenty:

A number is called oban if its name (in English) does not contain the letter "o".

Assuming that the name of every power of 10 greater than  $10^5$  ends in "-illion" (like million, billion, trillion, etc.), then the oban numbers are finite. There are 454 of them, the largest begin 999.

The numbers are as follows with Figure 4 providing a graph of these numbers. 

3, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 19, 20, 23, 25, 26, 27, 28, 29, 30, 33, 35, 36, 37, 38, 39, 50, 53, 55, 56, 57, 58, 59, 60, 63, 65, 66, 67, 68, 69, 70, 73, 75, 76, 77, 78, 79, 80, 83, 85, 86, 87, 88, 89, 90, 93, 95, 96, 97, 98, 99, 300, 303, 305, 306, 307, 308, 309, 310, 311, 312, 313, 315, 316, 317, 318, 319, 320, 323, 325, 326, 327, 328, 329, 330, 333, 335, 336, 337, 338, 339, 350, 353, 355, 356, 357, 358, 359, 360, 363, 365, 366, 367, 368, 369, 370, 373, 375, 376, 377, 378, 379, 380, 383, 385, 386, 387, 388, 389, 390, 393, 395, 396, 397, 398, 399, 500, 503, 505, 506, 507, 508, 509, 510, 511, 512, 513, 515, 516, 517, 518, 519, 520, 523, 525, 526, 527, 528, 529, 530, 533, 535, 536, 537, 538, 539, 550, 553, 555, 556, 557, 558, 559, 560, 563, 565, 566, 567, 568, 569, 570, 573, 575, 576, 577, 578, 579, 580, 583, 585, 586, 587, 588, 589, 590, 593, 595, 596, 597, 598, 599, 600, 603, 605, 606, 607, 608, 609, 610, 611, 612, 613, 615, 616, 617, 618, 619, 620, 623, 625, 626, 627, 628, 629, 630, 633, 635, 636, 637, 638, 639, 650, 653, 655, 656, 657, 658, 659, 660, 663, 665, 666, 667, 668, 669, 670, 673, 675, 676, 677, 678, 679, 680, 683, 685, 686, 687, 688, 689, 690, 693, 695, 696, 697, 698, 699, 700, 703, 705, 706, 707, 708, 709, 710, 711, 712, 713, 715, 716, 717, 718, 719, 720, 723, 725, 726, 727, 728, 729, 730, 733, 735, 736, 737, 738, 739, 750, 753, 755, 756, 757, 758, 759, 760, 763, 765, 766, 767, 768, 769, 770, 773, 775, 776, 777, 778, 779, 780, 783, 785, 786, 787, 788, 789, 790, 793, 795, 796, 797, 798, 799, 800, 803, 805, 806, 807, 808, 809, 810, 811, 812, 813, 815, 816, 817, 818, 819, 820, 823, 825, 826, 827, 828, 829, 830, 833, 835, 836, 837, 838, 839, 850, 853, 855, 856, 857, 858, 859, 860, 863, 865, 866, 867, 868, 869, 870, 873, 875, 876, 877, 878, 879, 880, 883, 885, 886, 887, 888, 889, 890, 893, 895, 896, 897, 898, 899, 900, 903, 905, 906, 907, 908, 909, 910, 911, 912, 913, 915, 916, 917, 918, 919, 920, 923, 925, 926, 927, 928, 929, 930, 933, 935, 936, 937, 938, 939, 950, 953, 955, 956, 957, 958, 959, 960, 963, 965, 966, 967, 968, 969, 970, 973, 975, 976, 977, 978, 979, 980, 983, 985, 986, 987, 988, 989, 990, 993, 995, 996, 997, 998, 999

Figure 4

Last come uban numbers defined by Numbers Aplenty as follows:

A number is called uban if its name (in English) does not contain the letter "u".

In particular, it cannot contain the terms "four", "hundred", and "thousand", So the uban number following 99 is 1000000. Despite being quite sparse, the sum of the reciprocals of uban numbers slowly diverges.

Here is a list of the initial uban numbers:

0, 1, 2, 3, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 32, 33, 35, 36, 37, 38, 39, 40, 41, 42, 43, 45, 46, 47, 48, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 75, 76, 77, 78, 79, 80, 81, 82, 83, 85, 86, 87, 88, 89, 90, 91, 92, 93, 95, 96, 97, 98, 99

Of course, you could choose the absence of certain consonants as well if you wanted to and the so-called tban numbers are in fact listed as OEIS A008523. The initial members of the sequence are:

0, 1, 4, 5, 6, 7, 9, 11, 100, 101, 104, 105, 106, 107, 109, 111, 400, 401, 404, 405, 406, 407, 409, 411, 500, 501, 504, 505, 506, 507, 509, 511, 600, 601, 604, 605, 606, 607, 609, 611, 700, 701, 704, 705, 706, 707, 709, 711, 900, 901, 904, 905, 906, 907, 909, 911, 1000000, 1000001, 1000004, 1000005 

That's probably enough as these types of numbers have no real mathematical significance but it was interesting to come across the idea of them and is relevant to my previous post on Numbers and Letters from July 21st 2023.