Aesthetic Numbers

An Aesthetic or Esthetic Number is defined by Numbers Aplenty as a number \(n\)  such that the difference between adjacent digits is 1 when the number is written in base \(b\). Such numbers attracted my attention because 27564, my diurnal age yesterday, is an aesthetic number in base 8 or simply 8-aesthetic because:$$27564_{10}=65654_8$$Let's look firstly at aesthetic numbers in base 10 between 10 and 40000. There are 149 such numbers and they are (permalink):

10, 12, 21, 23, 32, 34, 43, 45, 54, 56, 65, 67, 76, 78, 87, 89, 98, 101, 121, 123, 210, 212, 232, 234, 321, 323, 343, 345, 432, 434, 454, 456, 543, 545, 565, 567, 654, 656, 676, 678, 765, 767, 787, 789, 876, 878, 898, 987, 989, 1010, 1012, 1210, 1212, 1232, 1234, 2101, 2121, 2123, 2321, 2323, 2343, 2345, 3210, 3212, 3232, 3234, 3432, 3434, 3454, 3456, 4321, 4323, 4343, 4345, 4543, 4545, 4565, 4567, 5432, 5434, 5454, 5456, 5654, 5656, 5676, 5678, 6543, 6545, 6565, 6567, 6765, 6767, 6787, 6789, 7654, 7656, 7676, 7678, 7876, 7878, 7898, 8765, 8767, 8787, 8789, 8987, 8989, 9876, 9878, 9898, 10101, 10121, 10123, 12101, 12121, 12123, 12321, 12323, 12343, 12345, 21010, 21012, 21210, 21212, 21232, 21234, 23210, 23212, 23232, 23234, 23432, 23434, 23454, 23456, 32101, 32121, 32123, 32321, 32323, 32343, 32345, 34321, 34323, 34343, 34345, 34543, 34545, 34565, 34567

Here are the frequencies for base 10 numbers being aesthetic in smaller bases in the range from 10 to 40000:

• base 9 - 165 e.g. 37825 --> 56787
• base 8 - 171 e.g. 33436 --> 101234
• base 7 - 164 e.g. 36881 --> 212345
• base 6 - 188 e.g. 38657 --> 454545
• base 5 - 182 e.g. 36119 --> 2123434
• base 4 - 188 e.g. 39867 --> 21232323
• base 3 - 134 e.g. 36905 --> 1212121212
• base 2 - 12 e.g. 21845 --> 101010101010101

There are 17, 32, 61, 116, 222, 424, 813, 1556, and 2986 10-aesthetic numbers with 2, 3, 4, 5, 6, 7, 8, 9 and 10 digits respectively. Table 1 shows a screenshot of the Numbers Aplenty website featuring an interesting formula:

Table 1