# Undulating number

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 3 digits and A ≠ B. The first few such numbers are:

101, 121, 131, 141, 151, 161, 171, 181, 191, 202, 212, 232, 242, 252, 262, 272, 282, 292, 303, 313, 323, 343, 353, 363, 373, 383, 393, 404, 414, 424, 434, 454, 464, 474, 484, 494, ... (sequence A046075 in the OEIS)

For the full sequence of undulating numbers, see .

Some higher undulating numbers are: 6363, 80808, 1717171.

For any n ≥ 3, there are 9 × 9 = 81 non-trivial n-digit undulating numbers, since the first digit can have 9 values (it cannot be 0), and the second digit can have 9 values when it must be different from the first.

## Undulating prime

An undulating prime is an undulating number that is also prime. In every base, all undulating primes having at least 3 digits have an odd number of digits. The undulating primes in base 10 are:

2, 3, 5, 7, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, 797, 919, 929, 18181, 32323, 35353, 72727, 74747, 78787, 94949, 95959, ... (sequence A032758 in the OEIS)

## References

• Weisstein, Eric W. "Undulating number". MathWorld.