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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 J. Browkin and 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.

MSC:

11A25 Arithmetic functions; related numbers; inversion formulas
11B25 Arithmetic progressions

Citations:

Zbl 0820.11003
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