A number is a canyon number in base    if its digits first decrease and then increase in that base, there is only one occurrence of the smallest digit, and the first and last digits are the same.

For example, 212, 7317, and 85234568 are all canyon numbers in base 10.

Here, I consider generalized canyon numbers in base 10, where the first and last digits may differ, as in 102, 9317, and 8523456. For brevity, I simply refer to them as canyon numbers.

In base    there are

canyon numbers. For base    the formula above gives a total of 347489 canyon numbers, from 101 to 9876543210123456789. Among these, there are 24356 primes (from 101 to 98765432101456789) but only 105 powerful numbers, from    to  .

The first canyon numbers (in base 10) are 101, 102, 103, 104, 105, 106, 107, 108, 109, 201, 202, 203, 204, 205, 206, 207, 208, 209, 212, 213, 214, 215, 216, 217, 218, 219, 301, 302 more terms

Below, the spiral pattern of canyon numbers up to 10,000. See the page on prime numbers for an explanation and links to similar pictures.