Piatetski-Shapiro sequences

By R. Baker, William D. Banks, Jorg Brudern, Igor E. Shparlinski, Andreas J. Weingartner. 2012

51 citations recorded by the source corpus.

We consider various arithmetic questions for the Piatetski-Shapiro sequences bncc (n = 1, 2, 3, . . .) with c > 1, c 6∈ N. We exhibit a positive function θ(c) with the property that the largest prime factor of bncc exceeds nθ(c)−e infinitely often. For c ∈ (1, 149 87 ) we show that the counting function of natural numbers n 6 x for which bncc is squarefree satisfies the expected asymptotic formula. For c ∈ (1, 147 145) we show that there are infinitely many Carmichael numbers composed entirely of primes of the form p = bncc. 2010 MSC Numbers: 11N25, 11L07.

Subfields: Algebra and Number Theory, Artificial Intelligence

View Paper: https://www.semanticscholar.org/paper/d6c48aa2e89159175dd24eb95ac65ad54f36a458

Citations in the corpus

• On a sum involving the Euler function (2018, unknown): “with Beatty ⌊αn+ β⌋ sequences, see, for example, [1,3,6,12,13], and Piatetski–Shapiro ⌊nγ⌋ sequences, see, for example, [2,4,5,7,14,15], with real α, β and γ.”