Carmichael numbers composed of Piatetski-Shapiro primes in Beatty sequences

Abstract The Piatetski-Shapiro sequences are sequences of the form <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msubsup> <m:mrow> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mrow> <m:mo stretchy="false">⌊</m:mo> <m:mrow> <m:msup> <m:mrow> <m:mi>n</m:mi> </m:mrow> <m:mrow> <m:mi>c</m:mi> </m:mrow> </m:msup> </m:mrow> <m:mo stretchy="false">⌋</m:mo> </m:mrow> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mrow> <m:mi>n</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:mi>∞</m:mi> </m:mrow> </m:msubsup> </m:math> {\left(\lfloor {n}^{c}\rfloor )}_{n=1}^{\infty } and the Beatty sequence is the sequence of integers <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msubsup> <m:mrow> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mrow> <m:mo stretchy="false">⌊</m:mo> <m:mrow> <m:mi>α</m:mi> <m:mi>n</m:mi> <m:mo>+</m:mo> <m:mi>β</m:mi> </m:mrow> <m:mo stretchy="false">⌋</m:mo> </m:mrow> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mrow> <m:mi>n</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:mi>∞</m:mi> </m:mrow> </m:msubsup> </m:math> {(\lfloor \alpha n+\beta \rfloor )}_{n=1}^{\infty } . We prove that there are infinitely many Carmichael numbers composed of entirely the primes from the intersection of a Piatetski-Shapiro sequence and a Beatty sequence for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>c</m:mi> <m:mo>∈</m:mo> <m:mfenced open="(" close=")"> <m:mrow> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mfrac> <m:mrow> <m:mn>19137</m:mn> </m:mrow> <m:mrow> <m:mn>18746</m:mn> </m:mrow> </m:mfrac> </m:mrow> </m:mfenced> </m:math> c\in \left(\phantom{\rule[-0.75em]{}{0ex}},1,\frac{19137}{18746}\right) , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>α</m:mi> <m:mo>&gt;</m:mo> <m:mn>1</m:mn> </m:math> \alpha \gt 1 irrational and of finite type by investigating the Piatetski-Shapiro primes in arithmetic progressions in a Beatty sequence. Moreover, we also discuss the intersection of a Piatetski-Shapiro sequence and multiple Beatty sequences in arithmetic progressions.