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undulating numbers
A number is undulating in base  $b$  if it has at least 3 digits and it is made of exactly two distict digits which alternate, like 252 or 373737 in base 10 or 21=(10101)2.

David Moews has proved that there are only 4 undulating squares, namely 121, 484, 676, and 69696.

The first numbers which are undulating in at least 2 bases b <= 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

Below, the spiral pattern of undulating numbers up to  $100^2$. See the page on prime numbers for an explanation and links to similar pictures.

spiral pattern of undulating numbers

The first undulating numbers in base 10 are 101, 121, 131, 141, 151, 161, 171, 181, 191, 202, 212, 232, 242, 252, 262, 272, 282, 292, 303, 313 more terms

Undulating numbers can also be... (you may click on names or numbers and on + to get more values)

a-pointer 101 181 ABA 242 aban 101 121 131 + 989 abundant 252 272 282 + 48484848 admirable 282 464 474 + 89898 alt.fact. 101 alternating 101 121 141 + 989898989 amenable 101 121 141 + 989898989 apocalyptic 434 646 686 + 29292 arithmetic 101 131 141 + 9898989 balanced p. 373 binomial 171 252 595 + 575757 brilliant 121 323 737 + 38383 c.decagonal 101 151 c.heptagonal 1919191 c.nonagonal 595 5050 c.octagonal 121 c.pentagonal 141 181 c.square 181 313 545 + 6161 c.triangular 15151 cake 232 2626 Carol 959 Chen 101 131 181 + 1616161 congruent 101 141 151 + 9898989 constructible 272 cube 343 Cullen 161 Cunningham 101 242 323 + 232323 Curzon 141 393 414 + 161616161 cyclic 101 131 141 + 9595959 D-number 141 303 393 + 3737373 d-powerful 262 373 2626 + 43434 de Polignac 373 757 959 + 73737373 decagonal 232 2626 151515 deceptive 4141 13131313 29292929 + 101010101 deficient 101 121 131 + 9898989 dig.balanced 141 202 212 + 181818181 Duffinian 121 161 171 + 9797979 eban 4040 6060 economical 101 121 131 + 5656565 emirpimes 1313 1717 3131 + 9797 equidigital 101 121 131 + 5656565 esthetic 101 121 212 + 989898989898989 evil 101 141 202 + 989898989 fibodiv 323 646 969 Friedman 121 343 frugal 343 gapful 121 242 363 + 56565656565 Giuga 858 good prime 101 191 727 929 happy 262 313 383 + 8585858 Harshad 171 252 414 + 9090909090 heptagonal 616 4141 60606 hex 919 hexagonal 5151 hoax 202 424 454 + 25252525 Hogben 343 757 10101 Honaker 131 house 434 iban 101 121 141 + 747474 idoneal 232 inconsummate 161 272 383 + 959595 interprime 282 363 393 + 50505050 Jacobsthal 171 junction 101 202 212 + 49494949 Kaprekar 5050 7272 818181 + 85858585858585 Lehmer 595 949 4141 + 61616161 lucky 141 151 171 + 9696969 magic 505 magnanimous 101 modest 515 545 818 + 1919191919 Moran 171 1818 13131 + 73737 Motzkin 323 nonagonal 474 969 595959 nude 212 424 515 + 424242424 oban 303 313 323 + 989 odious 121 131 151 + 929292929 palindromic 101 121 131 + 989898989898989 palprime 101 131 151 + 737373737373737 pancake 121 191 232 panconsummate 121 353 pandigital 141 898 8585 + 434343 partition 101 pentagonal 1717 pernicious 121 131 151 + 9797979 power 121 343 484 + 69696 powerful 121 343 484 + 69696 practical 252 272 414 + 6969696 prim.abundant 272 282 464 + 89898 prime 101 131 151 + 91919191919 pronic 272 Proth 161 353 545 + 929 pseudoperfect 252 272 282 + 989898 repunit 121 343 585 + 101010101010101 Ruth-Aaron 949 5959 6868 19191919 self 121 323 424 + 797979797 semiprime 121 141 161 + 9797979 sliding 101 1010 Smith 121 202 454 + 90909090 sphenic 282 434 474 + 9898989 square 121 484 676 69696 star 121 181 strobogrammatic 101 181 808 + 818181818181818 strong prime 101 191 727 + 1616161 super Niven 1010 2020 3030 + 909090 super-d 131 181 454 + 9696969 tau 232 252 424 + 696969696 tetrahedral 969 triangular 171 595 5050 5151 truncatable prime 313 353 373 + 797 twin 101 151 181 + 1212121 Ulam 131 282 363 + 8282828 unprimeable 515 535 626 + 8989898 untouchable 262 292 474 + 969696 upside-down 1919 2828 3737 + 91919191919191 wasteful 171 202 212 + 9898989 weak prime 131 151 181 + 1212121 Woodall 191 323 383 Zuckerman 212 1212 13131 Zumkeller 252 272 282 + 89898