Highly totient number
Encyclopedia
A highly totient number k is an integer that has more solutions to the equation φ(x) = k, where φ is Euler's totient function
, than any integer below it. The first few highly totient numbers are
1, 2, 4, 8, 12
, 24
, 48
, 72
, 144
, 240
, 432, 480, 576, 720
, 1152, 1440 .
with 1, 3, 4, 5, 6, 10, 11, 17, 21, 31, 34, 37, 38, 49, 54, and 72 totient solutions respectively. The sequence of highly totient numbers is a subset of the sequence of smallest number k with exactly n solutions to φ(x) = k.
These numbers have more ways of being expressed as products of numbers of the form p - 1 and their products than smaller integers.
The concept is somewhat analogous to that of highly composite number
s, and in the same way that 1 is the only odd highly composite number, it is also the only odd highly totient number (indeed, the only odd number to not be a nontotient
). And just as there are infinitely many highly composite numbers, there are also infinitely many highly totient numbers, though the highly totient numbers get tougher to find the higher one goes, since calculating the totient function involves factorization into prime
s, something that becomes extremely difficult as the numbers get larger.
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...
, than any integer below it. The first few highly totient numbers are
1, 2, 4, 8, 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...
, 24
24 (number)
24 is the natural number following 23 and preceding 25.The SI prefix for 1024 is yotta , and for 10−24 yocto...
, 48
48 (number)
48 is the natural number following 47 and preceding 49. It is one third of a gross or four dozens.- In mathematics :Forty-eight is a double factorial of 6, a highly composite number. Like all other multiples of 6, it is a semiperfect number. 48 is the second 17-gonal number.48 is in abundance...
, 72
72 (number)
72 is the natural number following 71 and preceding 73. It is half a gross or 6 dozen .-In mathematics:...
, 144
144 (number)
144 is the natural number following 143 and preceding 145. 144 is a dozen dozens, or one gross.-In mathematics:It is the twelfth Fibonacci number, and the largest one to also be a square, as the square of 12 , following 89 and preceding 233.144 is the smallest number with exactly 15 divisors.144 is...
, 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...
, 432, 480, 576, 720
720 (number)
Seven hundred twenty is the natural number following 719 and preceding 721.It is 6! , a composite number with thirty divisors, more than any number below, making it a highly composite number...
, 1152, 1440 .
with 1, 3, 4, 5, 6, 10, 11, 17, 21, 31, 34, 37, 38, 49, 54, and 72 totient solutions respectively. The sequence of highly totient numbers is a subset of the sequence of smallest number k with exactly n solutions to φ(x) = k.
These numbers have more ways of being expressed as products of numbers of the form p - 1 and their products than smaller integers.
The concept is somewhat analogous to that of highly composite number
Highly composite number
A highly composite number is a positive integer with more divisors than any positive integer smaller than itself.The initial or smallest twenty-one highly composite numbers are listed in the table at right....
s, and in the same way that 1 is the only odd highly composite number, it is also the only odd highly totient number (indeed, the only odd number to not be a nontotient
Nontotient
In number theory, a nontotient is a positive integer n which is not in the range of Euler's totient function φ, that is, for which φ = n has no solution. 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,...
). And just as there are infinitely many highly composite numbers, there are also infinitely many highly totient numbers, though the highly totient numbers get tougher to find the higher one goes, since calculating the totient function involves factorization into prime
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
s, something that becomes extremely difficult as the numbers get larger.