Undulating Number

A number of the form $aba\cdots$, $abab\cdots$, etc. The first few nontrivial undulants (with the stipulation that $a\not=b$) are 101, 121, 131, 141, 151, 161, 171, 181, 191, 202, 212, ... (Sloane's A046075). Including the trivial 1- and 2-digit undulants and dropping the requirement that $a\not=b$ gives Sloane's A033619.

The first few undulating Squares are 121, 484, 676, 69696, ... (Sloane's A016073), with no larger such numbers of fewer than a million digits (Pickover 1995). Several tricks can be used to speed the search for square undulating numbers, especially by examining the possible patterns of ending digits. For example, the only possible sets of four trailing digits for undulating Squares are 0404, 1616, 2121, 2929, 3636, 6161, 6464, 6969, 8484, and 9696.

The only undulating Power $n^p=aba\cdots$ for $3\leq p\leq 31$ and up to 100 digits is $7^3=343$ (Pickover 1995). A large undulating prime is given by $7+720(100^{49}-1)/99$ (Pickover 1995).

A binary undulant is a Power of 2 whose base-10 representation contains one or both of the sequences $010\cdots$ and $101\cdots$. The first few are $2^n$ for $n=103$, 107, 138, 159, 179, 187, 192, 199, 205, ... (Sloane's A046076). The smallest $n$ for which an undulating sequence of exactly $d$-digit occurs for $d=3$, 4, ... are $n=103$, 138, 875, 949, 6617, 1802, 14545, ... (Sloane's A046077). An undulating binary sequence of length 10 occurs for $n=1,748,219$ (Pickover 1995).

References

Pickover, C. A. ``Is There a Double Smoothly Undulating Integer?'' In Computers, Pattern, Chaos and Beauty. New York: St. Martin's Press, 1990.

Pickover, C. A. ``The Undulation of the Monks.'' Ch. 20 in Keys to Infinity. New York: W. H. Freeman, pp. 159-161 1995.

Sloane, N. J. A. Sequences A016073, A033619, A046075, A046076, and A046077 in ``An On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html.