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metadromes
A number is a metadrome in a given base  $b$  (often 10 or 16) if its digits are in strictly increasing order in that base.

For example, 1234, 68 and 12789 are all metadromes in base 10.

If we allow the digits of a metadrome to be non-strictly increasing (i.e., nondecreasing, like in 1334558 or 2222, we obtain the plaindromes.

Similarly, the numbers whose digits are nonincreasing and strictly increasing are called nialpdromes and katadromes, respectively.

The total number metadromes in base  $b$  is equal to  $2^{b-1}$, hence in base 10 there are  $2^{10-1} = 512$  metadromes, from 0 to 123456789.

 $p_{13479}=145679$  is the largest metadromic prime whose index is a metadrome too.

The first metadromes (in base 10) are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 13, 14, 15, 16, 17, 18, 19, 23, 24, 25, 26, 27, 28, 29, 34 more terms

Pictorial representation of remainders (mod 2, 3, ...,11) frequency. For a table of values and more details click here
remainders of metadromes

A graph displaying how many metadromes are multiples of the primes p from 2 to 71. In black the ideal line 1/p.
divisibility of metadromes
Useful links Mathworld, Metadrome
OEIS, Sequence A009993 (base 10)
OEIS, Sequence A023784 (base 16)
Metadromes can also be... (you may click on names or numbers and on + to get more values)

a-pointer 13 3469 123457 ABA 18 24 128 + 1568 aban 12 13 14 + 789 abundant 12 18 24 + 12345678 Achilles 1568 3456 12348 admirable 12 24 56 + 1234578 alt.fact. 19 alternating 12 14 16 + 123456789 amenable 12 13 16 + 123456789 apocalyptic 157 245 247 + 26789 arithmetic 13 14 15 + 3456789 astonishing 15 27 automorphic 25 balanced p. 157 257 1367 + 12356789 Bell 15 betrothed 48 binomial 15 28 35 + 2346 brilliant 14 15 25 + 234679 c.heptagonal 148 2458 c.nonagonal 28 136 1378 c.octagonal 25 49 169 + 134689 c.pentagonal 16 456 c.square 13 25 145 c.triangular 19 46 136 + 4789 cake 15 26 378 23479 Canada 125 Carol 47 Catalan 14 Chen 13 17 19 + 23456789 compositorial 24 congruent 13 14 15 + 3456789 constructible 12 15 16 + 257 cube 27 125 Cullen 25 Cunningham 15 17 24 + 13457 Curzon 14 18 26 + 12345678 cyclic 13 15 17 + 2356789 D-number 15 39 57 + 2346789 d-powerful 24 89 135 + 2456789 de Polignac 127 149 1259 + 23456789 decagonal 27 126 deceptive 259 12345679 deficient 13 14 15 + 3456789 dig.balanced 12 15 19 + 12345689 double fact. 15 48 Duffinian 16 25 27 + 2456789 eban 34 36 46 56 economical 13 14 15 + 13456789 emirp 13 17 37 + 1235789 emirpimes 15 26 39 + 12345789 enlightened 256 equidigital 13 14 15 + 13456789 eRAP 24 esthetic 12 23 34 + 123456789 Eulerian 26 57 247 evil 12 15 17 + 123456789 factorial 24 fibodiv 14 19 28 + 12356 Fibonacci 13 34 89 Friedman 25 125 126 + 235678 frugal 125 128 256 + 134689 gapful 135 1245 1278 + 12346789 Gilda 29 49 78 good prime 17 29 37 + 1245689 happy 13 19 23 + 3456789 harmonic 28 Harshad 12 18 24 + 12345679 heptagonal 18 34 148 + 1345789 hex 19 37 127 + 1234567 hexagonal 15 28 45 378 highly composite 12 24 36 48 hoax 58 136 234 + 12456789 Hogben 13 57 157 Honaker 457 12347 13457 house 78 hungry 17 hyperperfect 28 iban 12 14 17 + 12347 iccanobiF 13 39 124 idoneal 12 13 15 + 357 impolite 16 128 256 interprime 12 15 18 + 1345679 Jordan-Polya 12 16 24 + 3456 Kaprekar 45 Kynea 23 79 Lehmer 15 247 259 + 12346789 Leyland 17 57 145 368 lonely 23 Lucas 18 29 47 123 lucky 13 15 25 + 2345689 Lynch-Bell 12 15 24 + 1368 m-pointer 23 magic 15 34 369 1379 magnanimous 12 14 16 + 2489 modest 13 19 23 + 2678 Moran 18 27 45 + 12345679 Motzkin 127 nonagonal 24 46 nude 12 15 24 + 1368 O'Halloran 12 36 156 oban 12 13 15 + 789 odious 13 14 16 + 23456789 pancake 16 29 37 + 2347 panconsummate 12 14 15 + 267 pandigital 15 19 78 + 124689 partition 15 56 135 pentagonal 12 35 145 + 1247 perfect 28 pernicious 12 13 14 + 3456789 Perrin 12 17 29 + 367 Pierpont 13 17 19 + 3457 plaindrome 12 13 14 + 123456789 power 16 25 27 + 134689 powerful 16 25 27 + 134689 practical 12 16 18 + 1245678 prim.abundant 12 18 56 + 1234578 prime 13 17 19 + 23456789 primeval 13 37 137 + 12345679 pronic 12 56 156 Proth 13 17 25 + 13569 pseudoperfect 12 18 24 + 345678 repfigit 14 19 28 47 repunit 13 15 57 + 259 Rhonda 1568 Ruth-Aaron 15 16 24 + 369 self 345 356 367 + 12456789 semiprime 14 15 25 + 12345789 sliding 25 29 1258 15689 Smith 27 58 346 + 12456789 Sophie Germain 23 29 89 + 145679 sphenic 78 138 238 + 13456789 square 16 25 36 + 134689 star 13 37 straight-line 123 135 147 + 123456789 strobogrammatic 69 689 strong prime 17 29 37 + 1456789 super Niven 12 24 36 48 super-d 19 69 127 + 2346789 superabundant 12 24 36 48 tau 12 18 24 + 1234568 tetrahedral 35 56 tetranacci 15 29 56 triangular 15 28 36 + 2346 tribonacci 13 24 149 trimorphic 24 25 49 + 1249 truncatable prime 13 17 23 + 12347 twin 13 17 19 + 1245689 uban 12 13 15 + 89 Ulam 13 16 18 + 1234578 unprimeable 1268 1345 1346 + 2345678 untouchable 124 146 238 + 245678 upside-down 19 28 37 + 123456789 wasteful 12 18 24 + 3456789 weak prime 13 19 23 + 23456789 Woodall 17 23 159 Zuckerman 12 15 24 + 135 Zumkeller 12 24 28 + 45678