Coprimality of integers in Piatetski-Shapiro sequences

Watcharapon Pimsert; Teerapat Srichan; Pinthira Tangsupphathawat

Czechoslovak Mathematical Journal (2023)

  • Volume: 73, Issue: 1, page 197-212
  • ISSN: 0011-4642

Abstract

top
We use the estimation of the number of integers n such that ⌊ n c ⌋ belongs to an arithmetic progression to study the coprimality of integers in ℕ c = { ⌊ n c ⌋ } n ∈ ℕ , c > 1 , c ∉ ℕ .

How to cite

top

Pimsert, Watcharapon, Srichan, Teerapat, and Tangsupphathawat, Pinthira. "Coprimality of integers in Piatetski-Shapiro sequences." Czechoslovak Mathematical Journal 73.1 (2023): 197-212. <http://eudml.org/doc/299413>.

@article{Pimsert2023,
abstract = {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 $\mathbb \{N\}^c=\lbrace \lfloor n^c \rfloor \rbrace _\{n\in \mathbb \{N\}\}$, $c>1$, $c\notin \mathbb \{N\}$.},
author = {Pimsert, Watcharapon, Srichan, Teerapat, Tangsupphathawat, Pinthira},
journal = {Czechoslovak Mathematical Journal},
keywords = {greatest common divisor; natural density; Piatetski-Shapiro sequence},
language = {eng},
number = {1},
pages = {197-212},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Coprimality of integers in Piatetski-Shapiro sequences},
url = {http://eudml.org/doc/299413},
volume = {73},
year = {2023},
}

TY - JOUR
AU - Pimsert, Watcharapon
AU - Srichan, Teerapat
AU - Tangsupphathawat, Pinthira
TI - Coprimality of integers in Piatetski-Shapiro sequences
JO - Czechoslovak Mathematical Journal
PY - 2023
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 73
IS - 1
SP - 197
EP - 212
AB - 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 $\mathbb {N}^c=\lbrace \lfloor n^c \rfloor \rbrace _{n\in \mathbb {N}}$, $c>1$, $c\notin \mathbb {N}$.
LA - eng
KW - greatest common divisor; natural density; Piatetski-Shapiro sequence
UR - http://eudml.org/doc/299413
ER -

References

top
  1. Bergelson, V., Richter, F. K., 10.1007/978-3-319-55357-3_5, Number Theory: Diophantine Problems, Uniform Distribution and Applications Springer, Cham (2017), 109-135. (2017) Zbl1414.11083MR3676397DOI10.1007/978-3-319-55357-3_5
  2. Cesàro, E., 10.1007/BF02420800, Ann. Mat. (2) 13 (1885), 235-250 French 9999JFM99999 17.0144.05. (1885) DOI10.1007/BF02420800
  3. Cesàro, E., 10.1007/BF02420803, Ann. Mat. (2) 13 (1885), 291-294 French 9999JFM99999 17.0145.01. (1885) DOI10.1007/BF02420803
  4. Cohen, E., 10.1090/S0002-9939-1960-0111713-8, Proc. Am. Math. Soc. 11 (1960), 164-171. (1960) Zbl0096.02906MR0111713DOI10.1090/S0002-9939-1960-0111713-8
  5. Delmer, F., Deshouillers, J.-M., 10.1023/A:1022337727769, Period. Math. Hung. 45 (2002), 15-20. (2002) Zbl1026.11003MR1955189DOI10.1023/A:1022337727769
  6. Deshouillers, J.-M., A remark on cube-free numbers in Segal-Piatetski-Shapiro sequences, Hardy-Ramanujan J. 41 (2018), 127-132. (2018) Zbl1448.11055MR3935505
  7. Diaconis, P., Erdős, P., 10.1214/lnms/1196285379, A Festschrift for Herman Rubin Institute of Mathematical Statistics Lecture Notes - Monograph Series 45. Institute of Mathematical Statistics, Beachwood (2004), 56-61. (2004) Zbl1268.11139MR2126886DOI10.1214/lnms/1196285379
  8. Dirichlet, G. L., 10.1017/CBO9781139237345.007, Abh. König. Preuss. Akad. Wiss. (1849), 69-83 German. (1849) DOI10.1017/CBO9781139237345.007
  9. Fernández, J. L., Fernández, P., 10.1142/S1793042115500062, Int. J. Number Theory 11 (2015), 89-126. (2015) Zbl1322.11102MR3280945DOI10.1142/S1793042115500062
  10. Hardy, G. H., Wright, E. M., An Introduction to the Theory of Numbers, Oxford University Press, Oxford (2008). (2008) Zbl1159.11001MR2445243
  11. Lambek, J., Moser, L., 10.4153/CJM-1955-020-0, Can. J. Math. 7 (1955), 155-158. (1955) Zbl0064.27903MR0068563DOI10.4153/CJM-1955-020-0
  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. (1953) Zbl0053.02702MR0059302

NotesEmbed ?

top

You must be logged in to post comments.