Piatetski-Shapiro sequences

Abstract

We consider various arithmetic questions for the Piatetski-Shapiro sequences ⌊n c ⌋ (n = 1, 2, 3, . ..) with c > 1, c ∈ N. We exhibit a positive function θ(c) with the property that the largest prime factor of ⌊n c ⌋ exceeds n θ(c)-ε infinitely often.For c ∈ (1, 149 87 ) we show that the counting function of natural numbers nx for which ⌊n c ⌋ 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 = ⌊n c ⌋. MSC Numbers: 11N25

Locations

• Acta Arithmetica - View
• arXiv (Cornell University) - View - PDF