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aban numbers
A number  $n$  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.

Aban numbers belong to the same family of eban numbers, iban numbers, oban numbers, and uban numbers. All quite boring, if I may speak my mind...

As for the above sets, the definition is not really precise, since large numbers may have non-standard names.

The first aban numbers are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20 more terms

Aban numbers can also be... (you may click on names or numbers and on + to get more values)

a-pointer 11 13 101 + 9999000709 ABA 18 24 32 + 999879000192 abundant 12 18 20 + 50000000 Achilles 72 108 200 + 999879000192 admirable 12 20 24 + 99000966 alt.fact. 19 101 619 alternating 10 12 14 + 989 amenable 12 13 16 + 1000000000 amicable 220 284 169000448 + 722254000455 anti-perfect 244 285 apocalyptic 157 192 218 + 994 arithmetic 11 13 14 + 9000998 astonishing 15 27 204 + 429 automorphic 25 76 376 625 balanced p. 53 157 173 + 9999000689 Bell 15 52 203 877 bemirp 10601000101 10901000101 11060000101 + 199889000881 betrothed 48 75 140 + 2359000280 binomial 10 15 20 + 999949000753 brilliant 10 14 15 + 999000973 c.decagonal 11 31 61 + 998426000911 c.heptagonal 22 43 71 + 999495000547 c.nonagonal 10 28 55 + 999949000753 c.octagonal 25 49 81 + 999998000001 c.pentagonal 16 31 51 + 998846000476 c.square 13 25 41 + 995778000313 c.triangular 10 19 31 + 998188000939 cake 15 26 42 + 823031000090 Canada 125 581 Carmichael 561 31992000001 58728000241 + 530598000241 Carol 47 223 959 Catalan 14 42 132 429 Chen 11 13 17 + 99000953 compositorial 24 192 congruent 13 14 15 + 9000999 constructible 10 12 15 + 960 cube 27 64 125 + 1000000000000 Cullen 25 65 161 + 897 Cunningham 10 15 17 + 999998000002 Curzon 14 18 21 + 199000998 cyclic 11 13 15 + 9000997 D-number 15 21 33 + 7000989 d-powerful 24 43 63 + 7000895 de Polignac 127 149 251 + 99000971 decagonal 10 27 52 + 997552000012 deceptive 91 259 451 + 99233000641 deficient 10 11 13 + 9000999 dig.balanced 10 11 12 + 199000996 double fact. 15 48 105 + 945 droll 72 240 672 + 409600000000 Duffinian 16 21 25 + 9000999 eban 30 32 34 + 66066000066 economical 10 11 13 + 20000000 emirp 13 17 31 + 199000981 emirpimes 15 26 39 + 99000998 enlightened 250 256 25000000 + 256000000000 equidigital 10 11 13 + 19000999 eRAP 20 24 98 + 993833000160 esthetic 10 12 21 + 989 Eulerian 11 26 57 + 502 evil 10 12 15 + 1000000001 factorial 24 120 720 fibodiv 14 19 28 + 969 Fibonacci 13 21 34 + 987 Friedman 25 121 125 + 736 frugal 125 128 243 + 999000877 gapful 100 105 108 + 100000000000 Gilda 29 49 78 + 997 Giuga 30 858 good prime 11 17 29 + 181000679 happy 10 13 19 + 10000000 harmonic 28 140 270 + 62487000576 Harshad 10 12 18 + 10000000000 heptagonal 18 34 55 + 996926000998 hex 19 37 61 + 995074000207 hexagonal 15 28 45 + 999949000753 highly composite 12 24 36 + 840 hoax 22 58 84 + 99000985 Hogben 13 21 31 + 999999000001 Honaker 131 263 457 + 999000991 house 32 78 155 + 561123000943 hungry 17 74 144 hyperperfect 21 28 301 + 840465000181 iban 10 11 12 + 777 iccanobiF 13 39 124 + 836 idoneal 10 12 13 + 840 impolite 16 32 64 + 512 inconsummate 62 63 65 + 993 insolite 111 interprime 12 15 18 + 99000984 Jacobsthal 11 21 43 + 683 Jordan-Polya 12 16 24 + 960 junction 101 103 105 + 99000943 Kaprekar 45 55 99 + 667000333 katadrome 10 20 21 + 987 Kynea 23 79 287 Lehmer 15 51 85 + 998432000161 Leyland 17 32 54 + 20000000000 lonely 23 53 120 211 Lucas 11 18 29 + 843 lucky 13 15 21 + 9000985 Lynch-Bell 12 15 24 + 936 m-pointer 23 61 magic 15 34 65 + 928166000645 magnanimous 11 12 14 + 84448000009 metadrome 12 13 14 + 789 modest 13 19 23 + 1990000717 Moran 18 21 27 + 99000756 Motzkin 21 51 127 + 835 narcissistic 153 370 371 407 nialpdrome 10 11 20 + 1000000000000 nonagonal 24 46 75 + 999028000434 nude 11 12 15 + 999 O'Halloran 12 20 36 + 924 oban 10 11 12 + 999 octagonal 21 40 65 + 991898000133 odious 11 13 14 + 1000000000 Ormiston 1000213 1000231 8000413 + 1995000331 palindromic 11 22 33 + 999000000999 palprime 11 101 131 + 98600000689 pancake 11 16 22 + 999949000754 panconsummate 10 11 12 + 721 pandigital 11 15 19 + 380000948 partition 11 15 22 + 792 pentagonal 12 22 35 + 998835000651 perfect 28 496 pernicious 10 11 12 + 9000992 Perrin 10 12 17 + 853 Pierpont 13 17 19 + 769 plaindrome 11 12 13 + 999 Poulet 341 561 645 + 949275000833 power 16 25 27 + 1000000000000 powerful 16 25 27 + 1000000000000 practical 12 16 18 + 10000000 prim.abundant 12 18 20 + 99000966 prime 11 13 17 + 999999000961 primeval 13 37 107 + 137 primorial 30 210 pronic 12 20 30 + 999999000000 Proth 13 17 25 + 999424000001 pseudoperfect 12 18 20 + 1000000 rare 65 repdigit 11 22 33 + 999 repfigit 14 19 28 + 742 repunit 13 15 21 + 999999000001 Ruth-Aaron 15 16 24 + 999602000842 Saint-Exupery 60 480 780 + 994560000000 Sastry 183 328 528 715 self 20 31 42 + 1000000005 self-describing 22 semiprime 10 14 15 + 99000998 sliding 11 20 25 + 925000000000 Smith 22 27 58 + 99000990 sphenic 30 42 66 + 99000997 square 16 25 36 + 1000000000000 star 13 37 73 + 987129000937 straight-line 111 123 135 + 999 strobogrammatic 11 69 88 + 999000000666 strong prime 11 17 29 + 99000973 subfactorial 44 265 super Niven 10 12 20 + 48888000000 super-d 19 31 69 + 9000988 superabundant 12 24 36 + 840 tau 12 18 24 + 1000000000 taxicab 27000008 27000064 216000064 + 999960000832 tetrahedral 10 20 35 + 702569000685 tetranacci 15 29 56 + 773 triangular 10 15 21 + 999949000753 tribonacci 13 24 44 + 927 trimorphic 24 25 49 + 500000000001 truncatable prime 13 17 23 + 997 twin 11 13 17 + 999000493 uban 10 11 12 + 1000000000000 Ulam 11 13 16 + 10000241 undulating 101 121 131 + 989 unprimeable 200 204 206 + 10000000 untouchable 52 88 96 + 996 upside-down 19 28 37 + 951 vampire 1206000297 1255000198 1255000950 + 8199000297 wasteful 12 18 20 + 9000999 weak prime 13 19 23 + 99000991 weakly prime 186000169 696000421 762000461 + 99884000693 weird 70 836 Woodall 17 23 63 + 895 Zeisel 105 26000605 524307000169 753601000663 Zuckerman 11 12 15 + 816 Zumkeller 12 20 24 + 996 zygodrome 11 22 33 + 999999000999