Czechoslovak Mathematical Journal, Vol. 71, No. 4, pp. 1063-1070, 2021
On the distribution of $(k,r)$-integers in Piatetski-Shapiro sequences
Teerapat Srichan
Received May 13, 2020. Published online March 26, 2021.
Abstract: A natural number $n$ is said to be a $(k,r)$-integer if $n=a^kb$, where $k>r>1$ and $b$ is not divisible by the $r$th power of any prime. We study the distribution of such $(k,r)$-integers in the Piatetski-Shapiro sequence $\{\lfloor n^c \rfloor\}$ with $c>1$. As a corollary, we also obtain similar results for semi-$r$-free integers.
References: [1] X. Cao, W. Zhai: The distribution of square-free numbers of the form $[n^c]$. J. Théor. Nombres Bordx. 10 (1998), 287-299. DOI 10.5802/jtnb.229 | MR 1828246 | Zbl 0926.11066
[2] X. Cao, W. Zhai: On the distribution of square-free numbers of the form $[n^c]$. II. Acta Math. Sin., Chin. Ser. 51 (2008), 1187-1194 Chinese. MR 2490038 | Zbl 1174.11395
[3] J.-M. Deshouillers: A remark on cube-free numbers in Segal-Piatetski-Shapiro sequences. Hardy-Ramanujan J. 41 (2018), 127-132. MR 3935505 | Zbl 1448.11055
[4] A. Ivić: The Riemann Zeta-Function: The Theory of the Riemann Zeta-Function. A Wiley-Interscience Publication. John Wiley & Sons, New York (1985). MR 792089 | Zbl 0556.10026
[5] I. I. Piatetski-Shapiro: On the distribution of prime numbers in sequences of the form $[f(n)]$. Mat. Sb., N.Ser. 33 (1953), 559-566. (In Russian.) MR 0059302 | Zbl 0053.02702
[6] G. J. Rieger: Remark on a paper of Stux concerning squarefree numbers in non-linear sequences. Pac. J. Math. 78 (1978), 241-242. DOI 10.2140/pjm.1978.78.241 | MR 513296 | Zbl 0395.10049
[7] I. E. Stux: Distribution of squarefree integers in non-linear sequences. Pac. J. Math. 59 (1975), 577-584. DOI 10.2140/pjm.1975.59.577 | MR 387218 | Zbl 0297.10033
[8] M. V. Subbarao, D. Suryanarayana: On the order of the error function of the $(k,r)$-integers. J. Number Theory 6 (1974), 112-123. DOI 10.1016/0022-314X(74)90049-3 | MR 0335454 | Zbl 0277.10036
[9] M. Zhang, J. Li: Distribution of cube-free numbers with form $[n^c]$. Front. Math. China 12 (2017), 1515-1525. DOI 10.1007/s11464-017-0652-1 | MR 3722230 | Zbl 1418.11137
Affiliations: Teerapat Srichan, Department of Mathematics, Faculty of Science, Kasetsart University, PO Box 50 Phahon Yothin Rd, Chatuchak District, Bangkok, 10900 Thailand, e-mail: fscitrp@ku.ac.th
