In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a sparsely totient number is a certain kind of even natural number

Natural number

In mathematics, the natural numbers are the ordinary whole numbers used for counting and ordering . These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively...

. A natural number, n, is sparsely totient if for all m > n,

φ(m)>φ(n),

where φ is Euler's totient function

Euler's totient function

In number theory, the totient \varphi of a positive integer n is defined to be the number of positive integers less than or equal to n that are coprime to n In number theory, the totient \varphi(n) of a positive integer n is defined to be the number of positive integers less than or equal to n that...

. The first few sparsely totient numbers are:

2, 6, 12

12 (number)

12 is the natural number following 11 and preceding 13.The word "twelve" is the largest number with a single-morpheme name in English. Etymology suggests that "twelve" arises from the Germanic compound twalif "two-leftover", so a literal translation would yield "two remaining [after having ten...

, 18

18 (number)

18 is the natural number following 17 and preceding 19.In speech, the numbers 18 and 80 are sometimes confused. When carefully enunciated, they differ in which syllable is stressed: 18 vs 80 . However, in dates such as 1864, or when contrasting numbers in the teens, such as 17, 18, 19, the stress...

, 30

30 (number)

30 is the natural number following 29 and preceding 31.-In mathematics:30 is the sum of the first four squares, which makes it a square pyramidal number.It is a primorial and is the smallest Giuga number....

, 42

42 (number)

42 is the natural number immediately following 41 and directly preceding 43. The number has received considerable attention in popular culture as a result of its central appearance in The Hitchhiker's Guide to the Galaxy as the "Answer to the Ultimate Question of Life, the Universe, and...

, 60

60 (number)

60 is the natural number following 59 and preceding 61. Being three times twenty, 60 is called "three score" in some older literature.-In mathematics:...

, 66

66 (number)

66 is the natural number following 65 and preceding 67.Usages of this number include:-Mathematics:*66 is a sphenic number, a triangular number, a hexagonal number, and a semi-meandric number...

, 90

90 (number)

90 is the natural number preceded by 89 and followed by 91.- In mathematics :*Because 90 is the sum of its unitary divisors , it is a unitary perfect number, and because it is equal to the sum of a subset of its divisors, it is a semiperfect number. 90 is a pronic number. But it is also a...

, 120

120 (number)

120 is the natural number following 119 and preceding 121. 120 was known as "the great hundred", especially prior to the year 1700, from the Teutonic Hundert which equalled 120. The number 100, now known commonly as "one hundred" was then known as "the small hundred". It is also known as...

, 126

126 (number)

126 is the natural number following 125 and preceding 127.-In mathematics:One hundred [and] twenty-six is a pentagonal pyramidal number, and a decagonal number as well as a pentatope number....

, 150

150 (number)

150 is the natural number following 149 and preceding 151.- In mathematics :*150 is the sum of eight consecutive primes...

, 210

210 (number)

210 is the natural number following 209 and preceding 211.- In mathematics :210 is a composite number, an abundant number, and the product of the first four prime numbers , and thus a primorial...

, 240

240 (number)

240 is the natural number following 239 and preceding 241.-In mathematics:With 20 divisors total , more than any previous number, 240 is a highly composite number, and it is also a refactorable number or tau number, since it has 20 divisors and 20 divides 240...

, 270

270 (number)

270 is the natural number following 269 and preceding 271.-In mathematics:*270 is a harmonic divisor number*270 is the fourth number that is divisible by its average integer divisor...

, 330, 420

420 (number)

420 is the natural number following 419 and preceding 421.420 is the sum of four consecutive primes . 420 is a zero of the Mertens function and is sparsely totient....

, 462, 510, 630 .

For example, 18 is a sparsely totient number because φ(18) = 6, and any number m > 18 falls into at least one of the following classes:

1. m has a prime factor p ≥ 11, so φ(m) ≥ φ(11) = 10 > φ(18).
2. m is a multiple of 7 and m/7 ≥ 3, so φ(m) ≥ 2φ(7) = 12 > φ(18).
3. m is a multiple of 5 and m/5 ≥ 4, so φ(m) ≥ 2φ(5) = 8 > φ(18).
4. m is a multiple of 3 and m/3 ≥ 7, so φ(m) ≥ 4φ(3) = 8 > φ(18).
5. m is a power of 2 and m ≥ 32, so φ(m) ≥ φ(32) = 16 > φ(18).