Undulating number
In mathematics, an undulating number is a number that has the digit form ABABAB... when in the base 10 number system. It is sometimes restricted to non-trivial undulating numbers, which are required to have at least three digits and A ≠ B. The first few undulating numbers are:
For the full sequence of undulating numbers, see.
Some larger undulating numbers are: 1010, 80808, 171717, 989898989.
Properties
- There are infinitely many undulating numbers.
- For any n ≥ 3, there are 9 × 9 = 81 non-trivial n-digit undulating numbers, since the first digit can have 9 values, and the second digit can have 9 values when it must be different from the first.
- Every undulating number with even number of digits and at least four digits is composite, since: ABABAB...AB = 10101...01 × AB. For example, 171717 = 10101 × 17.
- Undulating numbers with odd number of digits are palindromic. They can be prime, for example 151.
- The undulating number ABAB...AB with n repetitions of AB can be expressed as AB × /99. For example, 171717 = 17 × /99.
- The undulating number ABAB...ABA with n repetitions of AB followed by one A can be expressed as /99. For example, 989898989 = /99
- Undulating numbers can be generalized to other bases. If a number in base with even number of digits is undulating, in base it is a repdigit.