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Infinite families of noncototients. (English) Zbl 0965.11003
Let as usual $\varphi$ denote Euler’s function. A positive integer $n$ is called a noncototient, if the equation $x- \varphi(x)= n$ has no solution $x$. The conjecture of P. Erdős and W. Sierpiński, that there are infinitely many such numbers, was confirmed by {\it J. Browkin} and {\it A. Schinzel} [Colloq. Math. 68, 55-58 (1995; Zbl 0820.11003)] by showing that every integer of the form $2^n\cdot k$ for $k= 509203$ is a noncototient. In the present paper the authors give sufficient conditions on positive integers $k$ with the same property. Further six other integers $k$ are determined.

11A25Arithmetic functions, etc.
11B25Arithmetic progressions
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