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Keywords:
$(k,r)$-integer; Piatetski-Shapiro sequence
$(k,r)$-integer; Piatetski-Shapiro sequence
Summary:
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] Cao, X., Zhai, W.: 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] Cao, X., Zhai, W.: 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] Deshouillers, J.-M.: A remark on cube-free numbers in Segal-Piatetski-Shapiro sequences. Hardy-Ramanujan J. 41 (2018), 127-132. MR 3935505 | Zbl 1448.11055
[4] Ivić, A.: 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] Piatetski-Shapiro, I. I.: On the distribution of prime numbers in sequences of the form $[f(n)]$. Mat. Sb., N.Ser. 33 (1953), 559-566 Russian. MR 0059302 | Zbl 0053.02702
[6] Rieger, G. J.: 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] Stux, I. E.: 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] Subbarao, M. V., Suryanarayana, D.: 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] Zhang, M., Li, J.: 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
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