Multiperfect numbers on lines of the Pascal triangle
type: Item
instance of: scholarly article
title: Multiperfect numbers on lines of the Pascal triangle
zbMATH Open document ID: 1171.11002
author: Florian Luca
author: Juan Luis Varona
published in: Journal of Number Theory
publication date: 2009-05-19T00:00:00Z
review text: A positive integer \(n\) is said to be \(k\)-fold perfect (or multiperfect or multiply perfect) if \(\sigma(n) = kn\). The 3-fold perfect numbers are also called triperfect, and only six of them are known: 120, 672, 523776, 459818240, 1476304896, 51001180160. Several multiperfect numbers are also known for every \(k\leq 11\). Their number varies from thousands for \(k = 8, 9, 10\), to only one for \(k = 11\), which has more than a thousand decimal digits and was discovered in 2001. Descartes discovered the first 4-fold number, and Fermat the first 5-fold number. Except for the well-known Euclid-Euler rule for \(k = 2\), no formula to generate multiperfect numbers is known. Lehmer proved that if \(n\) is odd, then \(n\) is perfect just if \(2n\) is 3-fold perfect. No odd multiperfect number is known. There are various extensions of perfect and multiperfect numbers in the literature. For example, a positive integer \(n\) is called \((m, k)\)-perfect if \(\sigma^{(m)}(n) =kn\), where \(\sigma^{(m)}\) denotes the \(m\)th fold iterate of the sum of divisors function \(\sigma\). The present authors study the multiperfect and \((2, k)\)-perfect numbers on straight lines through the Pascal triangle. They show that except for the lines parallel to the edges, all other lines through the Pascal triangle contain at most finitely many multiperfect and \((2, k)\)-perfect numbers. The authors also show, among others, that the only \(n\) such that \(B_n\) is multiperfect is \(n = 2\) and the only \(n\) such that \(C_n\) is multiperfect is \(n = 1\), where \(B_n={2n\choose n}\) is the \(n\)th middle binomial coefficient and \(C_n={1\over n+1} B_n\) is the \(n\)th Catalan number. For \((2, k)\)-perfect numbers these values are \(n = 1\) and \(n = 1, 2, 5\). Further, the authors study the distribution of the numbers \(\sigma(n)/n\) whenever the positive integer \(n\) ranges through the binomial coefficients on a fixed line through the Pascal triangle. For a general account of perfect …
reviewed by: Pentti Haukkanen
Mathematics Subject Classification ID: 11A25
Mathematics Subject Classification ID: 11B65
zbMATH DE Number: 5554701
zbMATH Keywords: Multiperfect
zbMATH Keywords: Multiply perfect
zbMATH Keywords: Perfect
zbMATH Keywords: Sum of divisors
zbMATH Keywords: Pascal triangle
zbMATH Keywords: Binomial coefficients
zbMATH Keywords: Catalan numbers
MaRDI profile type: Publication
full work available at URL: j.jnt.2008.10.003
OpenAlex ID: W2088347007
cites work: Q4040797
cites work: Explicit bounds on exponential sums and the scarcity of squarefree binomial coefficients
cites work: Unsolved problems in number theory
cites work: Perfect Fibonacci and Lucas numbers
cites work: The Anti-Social Fermat Number
cites work: Multiply perfect numbers in Lucas sequences with odd parameters
cites work: Q4680230
cites work: Q5313958
cites work: On some problems of Mąkowski-Schinzel and Erdős concerning the arithmetical functions \(\varphi\) and \(\sigma\)
cites work: Arithmetic functions with linear recurrence sequences
cites work: Approximate formulas for some functions of prime numbers
cites work: Q4432173
cites work: Q5721265
cites work: Some arithmetic properties of recurrence sequences
cites work: Q4158939
DOI: 10.1016/J.JNT.2008.10.003
description: scientific article; zbMATH DE number 5554701