Perfect totient number
Encyclopedia
In number theory
, a perfect totient number is an integer
that is equal to the sum of its iterated totients. That is, we apply the totient function to a number n, apply it again to the resulting totient, and so on, until the number 1 is reached, and add together the resulting sequence of numbers; if the sum equals n, then n is a perfect totient number. Or to put it algebraically, if
where
is the iterated totient function and c is the integer such that
then n is a perfect totient number.
The first few perfect totient numbers are
For example, start with 327. φ(327) = 216, φ(216) = 72, φ(72) = 24, φ(24) = 8, φ(8) = 4, φ(4) = 2, φ(2) = 1, and 216 + 72 + 24 + 8 + 4 + 2 + 1 = 327.
Venkataraman (1975) found another family of perfect totient numbers: if p = 4×3k+1 is prime, then 3p is a perfect totient number. The values of k leading to perfect totient numbers in this way are
More generally if p is a prime number
greater than three, and 3p is a perfect totient number, then p ≡ 1 (mod 4) (Mohan and Suryanarayana 1982). Not all p of this form lead to perfect totient numbers; for instance, 51 is not a perfect totient number. Iannucci et al. (2003) showed that if 9p is a perfect totient number then p is a prime of one of three specific forms listed in their paper. It is not known whether there are any perfect totient numbers that are multiples of powers of 3 greater than 9 but not themselves powers of three.
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
, a perfect totient number is an integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
that is equal to the sum of its iterated totients. That is, we apply the totient function to a number n, apply it again to the resulting totient, and so on, until the number 1 is reached, and add together the resulting sequence of numbers; if the sum equals n, then n is a perfect totient number. Or to put it algebraically, if
where
is the iterated totient function and c is the integer such that
then n is a perfect totient number.
The first few perfect totient numbers are
- 3, 9, 1515 (number)15 is the natural number following 14 and preceding 16. In English, it is the smallest natural number with seven letters in its spelled name....
, 2727 (number)27 is the natural number following 26 and preceding 28.- In mathematics :Twenty-seven is a perfect cube, being 33 = 3 × 3 × 3. 27 is also 23 . There are exactly 27 straight lines on a smooth cubic surface, which give a basis of the fundamental representation of the E6 Lie algebra...
, 3939 (number)39 is the natural number following 38 and preceding 40.- In mathematics :Thirty-nine is the sum of five consecutive primes and the sum of the first three powers of 3...
, 8181 (number)81 is the natural number following 80 and preceding 82.-In mathematics:Eighty-one is the square of 9 and the fourth power of 3. Like all powers of three, 81 is a perfect totient number. It is a heptagonal number and a centered octagonal number. It is also a tribonacci number, and an open meandric...
, 111111 (number)111 is the natural number following 110 and preceding 112. It is the lowest positive integer requiring six syllables to name in American English, or seven syllables in Canadian and British English...
, 183183 (number)183 is the natural number following 182 and preceding 184.-In mathematics:* 183 is an odd number* 183 is a central polygonal number, n^2 - n + 1, where n=14 * 183 is a composite number* 183 is a deficient number, as 65 is less than 183...
, 243243 (number)243 is the natural number following 242 and preceding 244.-In mathematics:243 is:*the largest 3-digit number that is a fifth power .*a perfect totient number....
, 255255 (number)255 is the natural number following 254 and preceding 256.-In mathematics:Its factorization makes it a sphenic number. Since 255 = 28 - 1, it is a Mersenne number, and the fourth such number not to be a prime number...
, 327, 363363 (number)363, three hundred [and] sixty three, is the integer after 362 and before 364.-In mathematics:* 363 is the sum of nine consecutive primes...
, 471, 729, 2187, 2199, 3063, 4359, 4375, ... .
For example, start with 327. φ(327) = 216, φ(216) = 72, φ(72) = 24, φ(24) = 8, φ(8) = 4, φ(4) = 2, φ(2) = 1, and 216 + 72 + 24 + 8 + 4 + 2 + 1 = 327.
Multiples and powers of three
It can be observed that many perfect totient are multiples of 3; in fact, 4375 is the smallest perfect totient number that is not divisible by 3. All powers of 3 are perfect totient numbers, as may be seen by induction using the fact thatVenkataraman (1975) found another family of perfect totient numbers: if p = 4×3k+1 is prime, then 3p is a perfect totient number. The values of k leading to perfect totient numbers in this way are
- 0, 1, 2, 3, 6, 14, 15, 39, 201, 249, 1005, 1254, 1635, ... .
More generally if p is a prime number
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...
greater than three, and 3p is a perfect totient number, then p ≡ 1 (mod 4) (Mohan and Suryanarayana 1982). Not all p of this form lead to perfect totient numbers; for instance, 51 is not a perfect totient number. Iannucci et al. (2003) showed that if 9p is a perfect totient number then p is a prime of one of three specific forms listed in their paper. It is not known whether there are any perfect totient numbers that are multiples of powers of 3 greater than 9 but not themselves powers of three.