Geodesic
Coprimality of integers in Piatetski-Shapiro sequences
Czechoslovak Mathematical Journal, Tome 73 (2023) no. 1, pp. 197-212 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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We use the estimation of the number of integers $n$ such that $\lfloor n^c \rfloor $ belongs to an arithmetic progression to study the coprimality of integers in $\mathbb {N}^c=\{ \lfloor n^c \rfloor \}_{n\in \mathbb {N}}$, $c>1$, $c\notin \mathbb {N}$.
We use the estimation of the number of integers $n$ such that $\lfloor n^c \rfloor $ belongs to an arithmetic progression to study the coprimality of integers in $\mathbb {N}^c=\{ \lfloor n^c \rfloor \}_{n\in \mathbb {N}}$, $c>1$, $c\notin \mathbb {N}$.
DOI : 10.21136/CMJ.2022.0044-22
Classification : 11A05, 11K06
Keywords: greatest common divisor; natural density; Piatetski-Shapiro sequence 
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Pimsert, Watcharapon; Srichan, Teerapat; Tangsupphathawat, Pinthira. Coprimality of integers in Piatetski-Shapiro sequences. Czechoslovak Mathematical Journal, Tome 73 (2023) no. 1, pp. 197-212. doi: 10.21136/CMJ.2022.0044-22

[1] Bergelson, V., Richter, F. K.: On the density of coprime tuples of the form $(n, \lfloor f_1(n) \rfloor, \dots ,\lfloor f_k(n) \rfloor)$, where $f_1, \dots , f_k$ are functions from a Hardy field. Number Theory: Diophantine Problems, Uniform Distribution and Applications Springer, Cham (2017), 109-135. | DOI | MR | JFM

[2] Cesàro, E.: Étude moyenne di plus grand commun diviseur de deux nombres. Ann. Mat. (2) 13 (1885), 235-250 French \99999JFM99999 17.0144.05. | DOI

[3] Cesàro, E.: Sur le plus grand commun diviseur de plusieurs nombres. Ann. Mat. (2) 13 (1885), 291-294 French \99999JFM99999 17.0145.01. | DOI

[4] Cohen, E.: Arithmetical functions of a greatest common divisor. I. Proc. Am. Math. Soc. 11 (1960), 164-171. | DOI | MR | JFM

[5] Delmer, F., Deshouillers, J.-M.: On the probability that $n$ and $[n^c]$ are coprime. Period. Math. Hung. 45 (2002), 15-20. | DOI | MR | JFM

[6] Deshouillers, J.-M.: A remark on cube-free numbers in Segal-Piatetski-Shapiro sequences. Hardy-Ramanujan J. 41 (2018), 127-132. | MR | JFM

[7] Diaconis, P., Erdős, P.: On the distribution of the greatest common divisor. A Festschrift for Herman Rubin Institute of Mathematical Statistics Lecture Notes - Monograph Series 45. Institute of Mathematical Statistics, Beachwood (2004), 56-61. | DOI | MR | JFM

[8] Dirichlet, G. L.: Über die Bestimmung der mittleren Werthe in der Zahlentheorie. Abh. König. Preuss. Akad. Wiss. (1849), 69-83 German. | DOI

[9] Fernández, J. L., Fernández, P.: Asymptotic normality and greatest common divisors. Int. J. Number Theory 11 (2015), 89-126. | DOI | MR | JFM

[10] Hardy, G. H., Wright, E. M.: An Introduction to the Theory of Numbers. Oxford University Press, Oxford (2008). | MR | JFM

[11] Lambek, J., Moser, L.: On integers $n$ relatively prime to $f(n)$. Can. J. Math. 7 (1955), 155-158. | DOI | MR | JFM

[12] Piatetski-Shapiro, I. I.: On the distribution of the prime numbers in sequences of the form $[f(n)]$. Mat. Sb., N. Ser. 33 (1953), 559-566. | MR | JFM

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