The totient function , also called Euler's totient function, is defined as the number of positive integers that are relatively prime to (i.e., do not contain any factor in common with) n, where 1 is counted as being relatively prime to all numbers. Since a number less than or equal to and relatively prime to a given number is called a totative, the totient function can be simply defined as the number of totatives of n. For example, there are eight totatives of 24 (1, 5, 7, 11, 13, 17, 19, and 23), so . The totient function is implemented in Mathematica as EulerPhi[n].

is always even for . By convention, , although Mathematica defines EulerPhi[0] equal to 0 for consistency with its FactorInteger[0] command. The first few values of for n = 1, 2, ... are 1, 1, 2, 2, 4, 2, 6, 4, 6, 4, 10, ... (Sloane's A000010). The totient function is given by the Möbius transform of 1, 2, 3, 4, ... (Sloane and Plouffe 1995, p. 22). is plotted above for small n.

For a prime p,

since all numbers less than p are relatively prime to p. If is a power of a prime, then the numbers that have a common factor with m are the multiples of p: p, , ..., . There are of these multiples, so the number of factors relatively prime to is

Now take a general m divisible by p. Let be the number of positive integers not divisible by p. As before, p, , ..., have common factors, so

Now let q be some other prime dividing m. The integers divisible by q are q, , ..., . But these duplicate pq, , ..., . So the number of terms that must be subtracted from to obtain is

and

By induction, the general case is then

An interesting identity relates to ,

Another identity relates the divisors d of n to n via

The totient function satisfies the inequality

for all n except n = 2 and n = 6 (Kendall and Osborn 1965; Mitrinovic and Sándor 1995, p. 9). Therefore, the only values of n for which are n = 3, 4, and 6. In addition, for composite n,

(Sierpinski and Schinzel 1988; Mitrinovic and Sándor 1995, p. 9).

also satisfies

where is the Euler-Mascheroni constant. The values of n for which are given by 3, 4, 5, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 22, ... (Sloane's A100966).

The divisor function satisfies the congruence

for all primes and no composite with the exception of 4, 6, and 22, where is the divisor function. This fact was proved by Subbarao (1974), despite the implication to the contrary, "is it true for infinitely many composite n?," stated in Guy (1994, p. 92). No composite solution is currently known to

(Honsberger 1976, p. 35).

The first few n for which

are given by 1, 3, 15, 104, 164, 194, 255, 495, 584, 975, ... (Sloane's A001274), which have common values , 2, 8, 48, 80, 96, 128, 240, 288, 480, ... (Sloane's A003275).

The only for which

is n = 5186, giving

(Guy 1994, p. 91).

Arithmetic progressions of n giving consecutive values include

	(18)
	(19)

(Guy 1994, p. 91). McCranie found an arithmetic progression of six numbers with equal totient functions,

	(20)

as well as other progressions of six numbers starting at 583200, 1166400, 1749600, ... (Sloane's A050518).

If the Goldbach conjecture is true, then for every positive integer m, there are primes p and q such that

(Guy 1994, p. 105). Erdos asked if this holds for p and q not necessarily prime, but this relaxed form remains unproven (Guy 2004, p. 160).

Guy (1994, p. 99) discussed solutions to

where is the divisor function. F. Helenius has found 365 such solutions, the first of which are 2, 8, 12, 128, 240, 720, 6912, 32768, 142560, 712800, ... (Sloane's A001229).

 

Dedekind Function, Euler's Totient Rule, Fermat's Little Theorem, Lehmer's Totient Problem, Leudesdorf Theorem, Noncototient, Nontotient, Silverman Constant, Totative, Totient Summatory Function, Totient Valence Function