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esthetic numbers
J.-M. De Koninck and N. Doyon call a number  $n$   $q$-esthetic if the difference between adjacent digits is 1 when the number is written in base  $q$. For brevity, I omit the base when it is equal to 10.

For example, 678989 is esthetic and  $812=(3432)_6$  is  $6$-esthetic.

There are 17, 32, 61, 116, 222, 424, 813, 1556, and 2986 esthetic numbers with  $2,3,\dots,10$  digits.

In general, De Koninck & Doyon have proved that the number  $N_q(r)$  of the  $q$-esthetic numbers of  $r$  digits is equal to

\[ N_q(r)=\frac{2^r}{q+1}\sum_{\substack{k=1,\ k\ {\mathrm odd}\\[1mm]k\neq(q+1)/2}}^q% \frac{\cos^r\alpha_k\cdot\sin^2\alpha_k}{(1-\cos\alpha_k)^2}\,, \]
where  $\alpha_k = \pi k/(q+1)$.

For some small values of  $q$  the expression above can be simplified. For instance, we have  $N_4(r)=F_{r+3}$  and

\[ \begin{array}{rcl} N_3(r) &\!\!\!=\!\!\!& 2^{r/2-2} ((3-2 \sqrt{2})(-1)^r+3+2 \sqrt{2})\,,\\ N_5(r) &\!\!\!=\!\!\!& 3^{r/2-1} ((7-4\sqrt{3}) (-1)^r+7+4 \sqrt{3})/2\,. \end{array} \]
When  $q=10$  it is not possible to obtain a closed formula, but the results of De Koninck & Doyon suggest a good approximation, i.e.,  $N_{10}(r)\approx 4.39765\cdot1.918986^r$.

The first esthetic numbers (I disregard those smaller than 10) are 10, 12, 21, 23, 32, 34, 43, 45, 54, 56, 65, 67, 76, 78, 87, 89, 98, 101 more terms

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

A graph displaying how many esthetic numbers are multiples of the primes p from 2 to 71. In black the ideal line 1/p.
divisibility of esthetic numbers
Useful links
Esthetic numbers can also be... (you may click on names or numbers and on + to get more values)

a-pointer 101 21212101 23210101 ABA 32 98 aban 10 12 21 + 989 abundant 12 54 56 + 43454565 Achilles 432 3456 admirable 12 54 56 + 98765454 alt.fact. 101 alternating 10 12 21 + 989898989 amenable 12 21 32 + 989898989 amicable 1210 apocalyptic 434 787 898 + 23456 arithmetic 21 23 43 + 9898989 automorphic 76 balanced p. 34543 432343 3212323 + 6789898987 binomial 10 21 45 + 5676765 brilliant 10 21 121 + 989878789 c.decagonal 101 2101 c.heptagonal 43 2101 c.nonagonal 10 c.octagonal 121 12321 1234321 + 123456787654321 c.pentagonal 76 456 c.square 545 c.triangular 10 45676 cake 232 Chen 23 67 89 + 67678789 congruent 21 23 34 + 9898989 constructible 10 12 32 34 cube 343 Cullen 65 Cunningham 10 65 101 + 123456765432101 Curzon 21 54 65 + 123434565 cyclic 23 43 65 + 9898789 D-number 21 87 123 + 5454321 d-powerful 43 89 1234 + 876543 de Polignac 4543 4567 8789 + 89878987 decagonal 10 232 deceptive 23212321 101010101 deficient 10 21 23 + 9898989 dig.balanced 10 12 21 + 65676565 Duffinian 21 32 65 + 9898987 eban 32 34 54 56 economical 10 21 23 + 12343454 emirp 12323 32321 1210123 + 121232101 emirpimes 123 321 1234 + 89876543 equidigital 10 21 23 + 12343454 eRAP 98 4345 evil 10 12 23 + 989898989 fibodiv 323 Fibonacci 21 34 89 + 6765 Friedman 121 343 12101 + 765432 frugal 343 76545 5456565 + 656543232 gapful 121 1010 1210 + 98987676567 Gilda 78 65676 good prime 67 101 happy 10 23 32 + 9878767 Harshad 10 12 21 + 9898987876 heptagonal 34 hex 1234567 hexagonal 45 5676765 highly composite 12 hoax 234 454 456 + 76545654 Hogben 21 43 343 10101 Honaker 34543 3212123 343456543 565454543 house 32 78 434 hyperperfect 21 iban 10 12 21 + 343212 idoneal 10 12 21 + 345 impolite 32 inconsummate 65 432 543 + 898787 interprime 12 21 34 + 98767878 Jacobsthal 21 43 Jordan-Polya 12 32 432 3456 junction 101 210 212 + 89898765 Kaprekar 45 katadrome 10 21 32 + 9876543210 Kynea 23 Lehmer 212323 Leyland 32 54 lonely 23 Lucas 76 123 lucky 21 43 67 + 9876787 Lynch-Bell 12 432 m-pointer 23 magic 34 65 magnanimous 12 21 23 + 101 metadrome 12 23 34 + 123456789 modest 23 89 545 + 1232121212 Moran 21 45 12345 + 98767678 Motzkin 21 323 nialpdrome 10 21 32 + 9876543210 nonagonal 123234 nude 12 212 432 + 434343432 O'Halloran 12 oban 10 12 23 + 989 octagonal 21 65 odious 21 32 56 + 989898789 palindromic 101 121 212 + 989898989898989 palprime 101 787 32323 + 989898787898989 pancake 56 67 121 + 2121234567676 panconsummate 10 12 21 + 121 pandigital 21 78 210 + 9876543210 partition 56 101 4565 pentagonal 12 210 3432 12345676565 pernicious 10 12 21 + 9898789 Perrin 10 12 persistent 9876543210 10123456789 89876543210 98765432101 plaindrome 12 23 34 + 123456789 power 32 121 343 + 1234567654321 powerful 32 121 343 + 123456787654321 practical 12 32 54 + 8765456 prim.abundant 12 56 78 + 98765454 prime 23 43 67 + 898989898987 primeval 10123 101234567 1012345678 10123456789 primorial 210 pronic 12 56 210 + 78987656 Proth 65 321 545 + 56565432321 pseudoperfect 12 54 56 + 989898 rare 65 repunit 21 43 121 + 101010101010101 Ruth-Aaron 78 self 121 323 345 + 989898789 semiprime 10 21 34 + 89878787 sliding 65 101 1010 Smith 121 454 654 + 98765432 Sophie Germain 23 89 12101 + 6787654343 sphenic 78 345 434 + 98987678 square 121 676 12321 + 123456787654321 star 121 straight-line 123 210 234 + 9876543210 strobogrammatic 101 10101 1010101 + 101010101010101 strong prime 67 101 787 + 89876767 super Niven 10 12 210 + 101010 super-d 454 5454 7656 + 9876545 superabundant 12 tau 12 56 232 + 898989876 tetrahedral 10 56 5456 6545 tetranacci 56 triangular 10 21 45 + 5676765 trimorphic 76 truncatable prime 23 43 67 twin 43 101 32321 + 345434567 uban 10 12 21 + 98 Ulam 87 434 456 + 8787898 undulating 101 121 212 + 989898989898989 unprimeable 898 3232 3234 + 9878765 untouchable 210 898 1212 + 987878 upside-down 456 654 34567 + 898987654321212 wasteful 12 34 45 + 9898989 weak prime 23 43 89 + 89876789 weird 343210 Woodall 23 323 Zeisel 1012121 Zuckerman 12 212 432 + 3232343232 Zumkeller 12 54 56 + 89898