On a result of Flammenkamp-Luca concerning noncototient sequence
The author finds some interesting properties of the Sierpiński sequence. He proves the following theorems: (I) Let \(p\) be an odd prime. The sequence \((2^{k}p), k=1,2, \dots\) is the Sierpiński sequence if and only if: 1) \(2p\) is a noncototient, 2) \(p\) is not a Mersenne prime, 3) \(2^{k}p-1\) is composite for every positive integer \(k \geq1\). (II) There are infinitely many primes \(p\) in the arithmetical progression \( m\prod q_{j}+p_{0}\), where \(p_{0} = 509203\), \(\prod q_{j} = 3\times 5 \times 7 \times 13 \times 17 \times 241 \) such that: 1) \(2^{k}p-1\) is composite for every positive integer \( k\geq1\), 2) \(p\) is not a Mersenne prime. (III) The number \(2p\), where \(p\) is an odd prime of the form \(n-\varphi (n)\) if and only if there are different odd primes \( p_{j}\), where \(j = 1,2,\dots,r\), \(r\geq1\) such that \(p = p_{1}p_{2}\dots p_{r}-\frac{1}{2}(p_{1}-1)(p_{2}-1)\dots(p_{r}-1)\).
scientific article; zbMATH DE number 5017538