An undulating number is a number that has the digit form ABABAB... when in the

base 10 The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numer ...

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 such numbers are: : 101, 121, 131,

141 141 may refer to: * 141 (number), an integer * AD 141, a year of the Julian calendar * 141 BC __NOTOC__ Year 141 BC was a year of the pre-Julian Roman calendar. At the time it was known as the Year of the Consulship of Caepio and Pompeius (or ...

, 151, 161,

171 Year 171 ( CLXXI) was a common year starting on Monday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Severus and Herennianus (or, less frequently, year 924 '' Ab urbe co ...

181 Year 181 ( CLXXXI) was a common year starting on Sunday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Aurelius and Burrus (or, less frequently, year 934 '' Ab urbe condi ...

191 Year 191 ( CXCI) was a common year starting on Friday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Apronianus and Bradua (or, less frequently, year 944 '' Ab urbe cond ...

202 Year 202 ( CCII) was a common year starting on Friday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Severus and Antoninus (or, less frequently, year 955 '' Ab urbe condi ...

212 Year 212 ( CCXII) was a leap year starting on Wednesday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Asper and Camilius (or, less frequently, year 965 '' Ab urbe condit ...

, 232,

242 Year 242 (Roman numerals, CCXLII) was a common year starting on Saturday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Gratus and Lepidus (or, less frequently, year 995 ...

252 Year 252 ( CCLII) was a leap year starting on Thursday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Trebonianus and Volusianus (or, less frequently, year 1005 '' Ab urb ...

262 __NOTOC__ Year 262 ( CCLXII) was a common year starting on Wednesday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Gallienus and Faustianus (or, less frequently, year ...

, 272,

282 Year 282 ( CCLXXXII) was a common year starting on Sunday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Probus and Victorinus (or, less frequently, year 1035 '' Ab urbe ...

292 __NOTOC__ Year 292 ( CCXCII) was a leap year starting on Friday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Hannibalianus and Asclepiodotus (or, less frequently, year ...

303 __NOTOC__ Year 303 ( CCCIII) was a common year starting on Friday (link will display the full calendar) of the Julian calendar. It was known in the Roman Empire as the Year of the Consulship of Diocletian and Maximian (or, less frequently, y ...

, 313, 323, 343, 353, 363, 373, 383, 393, 404, 414, 424, 434, 454, 464, 474, 484, 494, ... 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 (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 Composite or compositing may refer to: Materials * Composite material, a material that is made from several different substances ** Metal matrix composite, composed of metal and other parts ** Cermet, a composite of ceramic and metallic materials ...

, since: ABABAB...AB = 10101...01 × AB. For example, 171717 = 10101 × 17. * Undulating numbers with odd number of digits are

palindromic A palindrome is a word, number, phrase, or other sequence of symbols that reads the same backwards as forwards, such as the words ''madam'' or ''racecar'', the date and time ''11/11/11 11:11,'' and the sentence: "A man, a plan, a canal – Pana ...

. They can be

prime A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...

, for example 151. * The undulating number ABAB...AB with ''n'' repetitions of AB can be expressed as AB × (10^2''n'' − 1)/99. For example, 171717 = 17 × (10⁶ − 1)/99. * The undulating number ABAB...ABA with ''n'' repetitions of AB followed by one A can be expressed as (AB × 10^2''n''+1 − BA)/99. For example, 989898989 = (98 × 10⁹ − 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 In recreational mathematics, a repdigit or sometimes monodigit is a natural number composed of repeated instances of the same digit in a positional number system (often implicitly decimal). The word is a portmanteau of repeated and digit. Example ...

Undulating primes

An undulating prime is an undulating number that is also prime. In every base, all undulating primes having at least three digits have an odd number of digits and are palindromic primes. 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, ...

References

External links

* Base-dependent integer sequences {{numtheory-stub