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The summatory of reciprocal totient function is defined as ():= = ()Edmund Landau showed in 1900 that this function has the asymptotic behavior (+ ) + + ()where γ is the Euler–Mascheroni constant,
A totient number is a value of Euler's totient function: that is, an m for which there is at least one n for which φ(n) = m. The valency or multiplicity of a totient number m is the number of solutions to this equation. [40] A nontotient is a natural number which is not a totient number. Every odd integer exceeding 1 is trivially a nontotient.
In number theory, a perfect totient number is an integer that is equal to the sum of its iterated totients.That is, one applies the totient function to a number n, apply it again to the resulting totient, and so on, until the number 1 is reached, and adds together the resulting sequence of numbers; if the sum equals n, then n is a perfect totient number.
For example, if one starts with Euler's totient function φ, and repeatedly applies the transformation process, one obtains: φ the totient function; φ ∗ 1 = I, where I(n) = n is the identity function; I ∗ 1 = σ 1 = σ, the divisor function; If the starting function is the Möbius function itself, the list of functions is: μ, the Möbius ...
32 is the fifth power of two (), making it the first non-unitary fifth-power of the form where is prime. 32 is the totient summatory function over the first 10 integers, [1] and the smallest number with exactly 7 solutions for ().
A related function is the divisor summatory function, which, as the name implies, ... where () is Euler's totient function. Then, the roots of () = ...
An average order of φ(n), Euler's totient function of n, is 6n / π 2; An average order of r(n), the number of ways of expressing n as a sum of two squares, is π; The average order of representations of a natural number as a sum of three squares is 4πn / 3;
Thus, a highly totient number is a number that has more ways of being expressed as a product of this form than does any smaller number. The concept is somewhat analogous to that of highly composite numbers , 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 ...