J.-M. De Koninck and N. Doyon call a number -esthetic
if the difference between adjacent digits is 1 when the number is written
. For brevity, I omit the base when it is equal to 10.
For example, 678989 is esthetic and is -esthetic.
There are 17, 32, 61, 116, 222, 424, 813, 1556, and 2986 esthetic
numbers with digits.
In general, De Koninck & Doyon have proved that the number of the
-esthetic numbers of digits is equal to
For some small values of the expression above can be simplified.
For instance, we have and
it is not possible to obtain a closed formula,
but the results of De Koninck & Doyon suggest a good approximation, i.e.,
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
A graph displaying how many esthetic numbers are multiples of the primes p
from 2 to 71. In black the ideal line 1/p