# Square-full numbers in Piatetski–Shapiro sequences

Square-full numbers in Piatetski–Shapiro sequences A positive integer n is called square-full if for every prime p|n\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$p\vert n$$\end{document}, also p2|n\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$p^2\vert n$$\end{document}. Piatetski–Shapiro sequences (PS-sequences) are defined by Nc=(⌊nc⌋)n∈N,(c>1,c∉N),\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}\begin{aligned} {\mathbb {N}}^c=(\lfloor n^c \rfloor )_{n\in {\mathbb {N}}}, \quad (c>1, c\notin {\mathbb {N}}), \end{aligned}\end{document}where ⌊z⌋\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\lfloor z\rfloor$$\end{document} is the integer part of a real z. In this paper we investigate the distribution of square-full numbers in Piatetski–Shapiro sequences.Résumé Un entier positif n est appelé un nombre puissant si pour chaque nombre premier p qui divise n, on a p2∣n\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$p^2\mid n$$\end{document}. Les séquences de Piatetski–Shapiro (PS-séquences) sont définies par Nc=(⌊nc⌋)n∈N,(c>1,c∉N),\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}\begin{aligned} {\mathbb {N}}^c=(\lfloor n^c \rfloor )_{n\in {\mathbb {N}}}, \ \ \ \ (c>1, c\notin {\mathbb {N}}), \end{aligned}\end{document}oú ⌊z⌋\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\lfloor z\rfloor$$\end{document} est la partie entie entiére d’un reél z. Dans cet article, nous étudions la distribution des nombres puissants dans des séquences de Piatetski–Shapiro. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Annales mathématiques du Québec Springer Journals

# Square-full numbers in Piatetski–Shapiro sequences

, Volume 44 (2) – Oct 15, 2020
7 pages

/lp/springer-journals/square-full-numbers-in-piatetski-shapiro-sequences-LVANhNKs9U
Publisher
Springer Journals
ISSN
2195-4755
eISSN
2195-4763
DOI
10.1007/s40316-020-00132-8
Publisher site
See Article on Publisher Site

### Abstract

A positive integer n is called square-full if for every prime p|n\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$p\vert n$$\end{document}, also p2|n\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$p^2\vert n$$\end{document}. Piatetski–Shapiro sequences (PS-sequences) are defined by Nc=(⌊nc⌋)n∈N,(c>1,c∉N),\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}\begin{aligned} {\mathbb {N}}^c=(\lfloor n^c \rfloor )_{n\in {\mathbb {N}}}, \quad (c>1, c\notin {\mathbb {N}}), \end{aligned}\end{document}where ⌊z⌋\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\lfloor z\rfloor$$\end{document} is the integer part of a real z. In this paper we investigate the distribution of square-full numbers in Piatetski–Shapiro sequences.Résumé Un entier positif n est appelé un nombre puissant si pour chaque nombre premier p qui divise n, on a p2∣n\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$p^2\mid n$$\end{document}. Les séquences de Piatetski–Shapiro (PS-séquences) sont définies par Nc=(⌊nc⌋)n∈N,(c>1,c∉N),\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}\begin{aligned} {\mathbb {N}}^c=(\lfloor n^c \rfloor )_{n\in {\mathbb {N}}}, \ \ \ \ (c>1, c\notin {\mathbb {N}}), \end{aligned}\end{document}oú ⌊z⌋\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\lfloor z\rfloor$$\end{document} est la partie entie entiére d’un reél z. Dans cet article, nous étudions la distribution des nombres puissants dans des séquences de Piatetski–Shapiro.

### Journal

Annales mathématiques du QuébecSpringer Journals

Published: Oct 15, 2020