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 integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Math ...

, a nontotient is a positive integer ''n'' which is not a totient number: it is not in the range of

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 ...

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'' coprimes 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,

114 114 may refer to: *114 (number) *AD 114 *114 BC *114 (1st London) Army Engineer Regiment, Royal Engineers, an English military unit *114 (Antrim Artillery) Field Squadron, Royal Engineers, a Northern Irish military unit *114 (MBTA bus) *114 (New Je ...

118 118 may refer to: *118 (number) *AD 118 *118 BC *118 (TV series) *118 (film) *118 (Tees) Corps Engineer Regiment *118 (Tees) Field Squadron, Royal Engineers See also *11/8 (disambiguation) *Oganesson Oganesson is a synthetic chemical element wi ...

122 122 may refer to: *122 (number), a natural number * AD 122, a year in the 2nd century AD * 122 BC, a year in the 2nd century BC * ''122'' (film), a 2019 Egyptian psychological horror film *"One Twenty Two", a 2022 single by the American rock band Bo ...

, 124, 134, 142, 146,

152 Year 152 ( CLII) 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 Glabrio and Homullus (or, less frequently, year 905 ''Ab urbe condita'' ...

154 Year 154 ( CLIV) 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 Aurelius and Lateranus (or, less frequently, year 907 ''Ab urbe cond ...

158 Year 158 ( CLVIII) 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 Tertullus and Sacerdos (or, less frequently, year 911 '' Ab urbe ...

170 Year 170 ( CLXX) 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 Clarus and Cornelius (or, less frequently, year 923 ''Ab urbe condita ...

, 174,

182 Year 182 ( CLXXXII) 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 Sura and Rufus (or, less frequently, year 935 ''Ab urbe condita'') ...

, 186,

188 Year 188 (CLXXXVIII) was a leap year starting on Monday of the Julian calendar. At the time, it was known in the Roman Empire as the Year of the Consulship of Fuscianus and Silanus (or, less frequently, year 941 ''Ab urbe condita''). The denomi ...

194 Year 194 ( CXCIV) was a common year starting on Tuesday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Septimius and Septimius (or, less frequently, year 947 '' Ab urbe ...

202 Year 202 (Roman numerals, 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 '' ...

206 Year 206 ( CCVI) 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 Umbrius and Gavius (or, less frequently, year 959 ''Ab urbe condit ...

214 Year 214 ( CCXIV) 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 Messalla and Suetrius (or, less frequently, year 967 ''Ab urbe con ...

, 218,

230 Year 230 ( 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 983 ''Ab urbe c ...

234 __NOTOC__ Year 234 ( CCXXXIV) 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 Pupienus and Sulla (or, less frequently, year 987 '' ...

236 __NOTOC__ Year 236 ( CCXXXVI) 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 Verus and Africanus (or, less frequently, year 989 ''Ab ...

242 Year 242 ( 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 ''Ab urbe cond ...

, 244, 246, 248,

254 Year 254 ( CCLIV) 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 Valerianus and Gallienus (or, less frequently, year 1007 '' Ab urbe ...

258 Year 258 ( CCLVIII) 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 Tuscus and Bassus (or, less frequently, year 1011 ''Ab urbe condit ...

266 __NOTOC__ Year 266 ( CCLXVI) 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 Gallienus and Sabinillus (or, less frequently, year 1019 ...

274 Year 274 ( CCLXXIV) was a common 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 Aurelianus and Capitolinus (or, less frequently, year 1027 ''Ab ...

278 __NOTOC__ Year 278 ( CCLXXVIII) was a common year starting on Tuesday (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 Lupus (or, less frequently, year 1031 ''A ...

284 __NOTOC__ Year 284 ( CCLXXXIV) was a leap year starting on Tuesday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Carinus and Numerianus (or, less frequently, year 1037 ...

286 __NOTOC__ Year 286 ( CCLXXXVI) 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 Maximus and Aquilinus (or, less frequently, year 1039 ...

290 __NOTOC__ Year 290 ( CCXC) was a common year starting on Wednesday (link will display the full calendar) of the Julian calendar. In the Roman Empire, it was known as the Year of the Consulship of Valerius and Valerius (or, less frequently, ye ...

298 Year 298 (Roman numerals, CCXCVIII) 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 Faustus and Gallus (or, less frequently, year 10 ...

, ... 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 way ...

, 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 ''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 PlanetMath * * {{Classes of natural numbers Integer sequences