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Infinite families of noncototients. (English) Zbl 0965.11003

Let as usual ϕ denote Euler’s function. A positive integer n is called a noncototient, if the equation x-ϕ(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 ·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:
11A25Arithmetic functions, etc.
11B25Arithmetic progressions