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On nontotients. (English) Zbl 0772.11001
Let $\phi \left(x\right)$ be Euler’s totient function. If the equation $\phi \left(x\right)=n$ has no solution, then $n$ is called a nontotient. In this paper, the author proves that a nontotient can have an arbitrary divisor and the author gives two sorts of odd numbers such that for the odd number $k$ of the first sort ${2}^{\alpha }·k$ is a nontotient for a given positive integer $a$ while for the odd number $k$ of the second sort, ${2}^{\alpha }·k$ is a nontotient for arbitrary positive integer $a$.
##### MSC:
 11A25 Arithmetic functions, etc.
##### Keywords:
Euler’s phi-function; nontotient