Undulating Numbers

I've only ever made a passing reference to an undulating number in a post titled 26262: A Special Palindrome from February 26th 2021. This post will address that omission. Firstly, let's have Numbers Aplenty define what is meant by an undulating number:

A number is undulating in base \(b\) if it has at least \(3\) digits and it is made of exactly two distinct digits which alternate, like \(252\) or \(373737\) in base \(10\) or \(21=10101_2\).

Undulating numbers can be termed undulants and in base 10 they comprise OEIS A046075:

A046075

Nontrivial undulants; base 10 numbers >100 which are of the form \(aba, abab, ababa, \dots \) where \(a\) and \(b\) are not equal.

 The initial members of this sequence 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, 505, 515, 525, 535, 545, 565, 575, 585, 595, 606, 616, 626, 636, 646, 656

Wikipedia lists these properties of undulating numbers:

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 (it cannot be 0), 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: $$ababab \dots ab = 10101 \dots 01 \times ab\\ \text{ e.g. } 171717 = 10101 \times 17$$Undulating numbers with odd number of digits are palindromic. They can be prime, for example 151.

The undulating number \( abab \dots ab\) with \(n\) repetitions of \(ab\) can be expressed as: $$ ab \times \frac{10^{2n} − 1}{99} \\ \text{ e.g. } 171717 = 17 \times \frac{10^6 − 1}{99}$$The undulating number \(abab \dots aba\) with \( n\) repetitions of \(ab\) followed by one \(a\) can be expressed as$$ ab \times \frac{10^{2n+1} − ba}{99}\\  \text{ e.g. } 989898989 = 98 \times \frac{10^9 − 89}{99}$$Undulating numbers can be generalized to other bases. If a number in base \(b\) with even number of digits is undulating, in base \(b^{2} \) it is a repdigit.

There can be confusion about what constitutes an undulating number. For some, the only requirement is that the numbers alternate between up and down or down and up. For this reason the term smoothly undulating has been introduced as explained below:

Smoothly Undulating Palindromic Primes (or SUPP's for short) are numbers that are primes, palindromic in base 10, and the digits alternate, but why smooth one might ask! The smoothness was added to make a difference with the normal undulating numbers. The description for normal undulating numbers is that the next digits alternately go up and down (or down and up) but the absolute difference values between two adjacent digits may differ e.g. 906343609. In a smoothly undulating number the absolute difference values between two adjacent digits are always equal, therefore only two distinct digits can appear in the number e.g. 74747474747474747. Source.

The smoothly undulating primes begin:

101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, 797, 919, 929, 18181, 32323, 35353, 72727, 74747, 78787, 94949, 95959, ...

Apparently there are only  four undulating squares, namely \(121, 484, 676\) and \(69696\) corresponding to \(11^2,22^2,26^2\) and \(264^2\). See https://oeis.org/A016073.

The first numbers which are undulating in at least two bases \(b \leq 16\) are 10, 46, 50, 55, 67, 78, 85, 92, 98, 100, 104, 109, 119, 121, 130, 135, 136, 141, 145, 151, 154, 164, 166, 170, 178, 181, 182, 185, 191, 197, 200, ...

For example, \(10_{10}=1010_2=101_3\) and is thus undulating in bases 2 and 3.