number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777� ...

, a nontotient is a positive integer ''n'' which is not a

totient number In number theory, Euler's totient function counts the positive integers up to a given integer that are relatively prime to . It is written using the Greek letter phi as \varphi(n) or \phi(n), and may also be called Euler's phi function. In ot ...

: it is not in the

range Range may refer to: Geography * Range (geographic), a chain of hills or mountains; a somewhat linear, complex mountainous or hilly area (cordillera, sierra) ** Mountain range, a group of mountains bordered by lowlands * Range, a term used to i ...

Euler's totient function In number theory, Euler's totient function counts the positive integers up to a given integer that are relatively prime to . It is written using the Greek letter phi as \varphi(n) or \phi(n), and may also be called Euler's phi function. In ot ...

φ, that is, the equation φ(''x'') = ''n'' has no solution ''x''. In other words, ''n'' is a nontotient if there is no integer ''x'' that has exactly ''n''

coprime In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivale ...

s below it. All odd numbers are nontotients, except 1, since it has the solutions ''x'' = 1 and ''x'' = 2. The first few even nontotients are : 14, 26, 34, 38, 50, 62, 68, 74, 76, 86, 90, 94, 98, 114, 118, 122, 124, 134,

142 142 may refer to: * 142 (number), an integer * AD 142 Year 142 ( CXLII) was a common year starting on Sunday (link will display the full calendar) of the Julian calendar. At the time, it was known in the Roman Empire as the Year of the Consul ...

, 146, 152, 154, 158, 170,

174 Year 174 ( CLXXIV) 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 Gallus and Flaccus (or, less frequently, year 927 '' Ab urbe condi ...

, 182,

186 Year 186 ( CLXXXVI) 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 Aurelius and Glabrio (or, less frequently, year 939 ''Ab urbe co ...

, 188, 194, 202, 206, 214, 218,

230 Year 230 (Roman numerals, CCXXX) 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 Agricola and Clementinus (or, less frequently, year ...

, 234, 236, 242,

244 __NOTOC__ Year 244 (Roman numerals, CCXLIV) was a leap 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 Tiberius Pollenius Armenius Peregrinus, Arm ...

246 __NOTOC__ Year 246 ( CCXLVI) was a common year starting on Thursday (link will display the full calendar) of the Julian calendar, the 246th Year of the Common Era ( CE) and Anno Domini ( AD) designations, the 246th year of the 1st millennium, th ...

248 __NOTOC__ Year 248 ( CCXLVIII) was a leap 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 Philippus and Severus (or, less frequently, year 1001 '' ...

, 254, 258, 266, 274, 278, 284, 286, 290, 298, ... Least ''k'' such that the totient of ''k'' is ''n'' are (0 if no such ''k'' exists) :1, 3, 0, 5, 0, 7, 0, 15, 0, 11, 0, 13, 0, 0, 0, 17, 0, 19, 0, 25, 0, 23, 0, 35, 0, 0, 0, 29, 0, 31, 0, 51, 0, 0, 0, 37, 0, 0, 0, 41, 0, 43, 0, 69, 0, 47, 0, 65, 0, 0, 0, 53, 0, 81, 0, 87, 0, 59, 0, 61, 0, 0, 0, 85, 0, 67, 0, 0, 0, 71, 0, 73, ... Greatest ''k'' such that the totient of ''k'' is ''n'' are (0 if no such ''k'' exists) :2, 6, 0, 12, 0, 18, 0, 30, 0, 22, 0, 42, 0, 0, 0, 60, 0, 54, 0, 66, 0, 46, 0, 90, 0, 0, 0, 58, 0, 62, 0, 120, 0, 0, 0, 126, 0, 0, 0, 150, 0, 98, 0, 138, 0, 94, 0, 210, 0, 0, 0, 106, 0, 162, 0, 174, 0, 118, 0, 198, 0, 0, 0, 240, 0, 134, 0, 0, 0, 142, 0, 270, ... Number of ''k''s such that φ(''k'') = ''n'' are (start with ''n'' = 0) :0, 2, 3, 0, 4, 0, 4, 0, 5, 0, 2, 0, 6, 0, 0, 0, 6, 0, 4, 0, 5, 0, 2, 0, 10, 0, 0, 0, 2, 0, 2, 0, 7, 0, 0, 0, 8, 0, 0, 0, 9, 0, 4, 0, 3, 0, 2, 0, 11, 0, 0, 0, 2, 0, 2, 0, 3, 0, 2, 0, 9, 0, 0, 0, 8, 0, 2, 0, 0, 0, 2, 0, 17, ... According to Carmichael's conjecture there are no 1's in this sequence. An even nontotient may be one more than a

prime number 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 ...

, but never one less, since all numbers below a prime number are, by definition, coprime to it. To put it algebraically, for p prime: φ(''p'') = ''p'' − 1. Also, a

pronic number A pronic number is a number that is the product of two consecutive integers, that is, a number of the form n(n+1).. The study of these numbers dates back to Aristotle. They are also called oblong numbers, heteromecic numbers,. or rectangular number ...

''n''(''n'' − 1) is certainly not a nontotient if ''n'' is prime since φ(''p''²) = ''p''(''p'' − 1). If a natural number ''n'' is a totient, it can be shown that ''n'' · 2^''k'' is a totient for all natural number ''k''. There are infinitely many even nontotient numbers: indeed, there are infinitely many distinct primes ''p'' (such as 78557 and 271129, see Sierpinski number) such that all numbers of the form 2^''a''''p'' are nontotient, and every odd number has an even multiple which is a nontotient.

References

* * L. Havelock
A Few Observations on Totient and Cototient Valence
from

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* * {{Classes of natural numbers Integer sequences