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:2756410=656548
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
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