Some results concerning exponential divisors

If the natural number n has the canonical form p1a1p2a2…prar then d=p1b1p2b2…prbr is said to be an exponential divisor of n if bi|ai for i=1,2,…,r. The sum of the exponential divisors of n is denoted by σ(e)(n). n is said to be an e-perfect number if σ(e)(n)=2n; (m;n) is said to be an e-amicable pai...

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Main Author: Peter Hagis
Format: Article
Language:English
Published: Wiley 1988-01-01
Series:International Journal of Mathematics and Mathematical Sciences
Subjects:
Online Access:http://dx.doi.org/10.1155/S0161171288000407
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author Peter Hagis
author_facet Peter Hagis
author_sort Peter Hagis
collection DOAJ
description If the natural number n has the canonical form p1a1p2a2…prar then d=p1b1p2b2…prbr is said to be an exponential divisor of n if bi|ai for i=1,2,…,r. The sum of the exponential divisors of n is denoted by σ(e)(n). n is said to be an e-perfect number if σ(e)(n)=2n; (m;n) is said to be an e-amicable pair if σ(e)(m)=m+n=σ(e)(n); n0,n1,n2,… is said to be an e-aliquot sequence if ni+1=σ(e)(ni)−ni. Among the results established in this paper are: the density of the e-perfect numbers is .0087; each of the first 10,000,000e-aliquot sequences is bounded.
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spelling doaj-art-86ac1c53483e42ffb832627a0c87e8232025-02-03T01:03:15ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04251988-01-0111234334910.1155/S0161171288000407Some results concerning exponential divisorsPeter Hagis0Mathematics Department, Temple University, Philadelphia 19122, PA, USAIf the natural number n has the canonical form p1a1p2a2…prar then d=p1b1p2b2…prbr is said to be an exponential divisor of n if bi|ai for i=1,2,…,r. The sum of the exponential divisors of n is denoted by σ(e)(n). n is said to be an e-perfect number if σ(e)(n)=2n; (m;n) is said to be an e-amicable pair if σ(e)(m)=m+n=σ(e)(n); n0,n1,n2,… is said to be an e-aliquot sequence if ni+1=σ(e)(ni)−ni. Among the results established in this paper are: the density of the e-perfect numbers is .0087; each of the first 10,000,000e-aliquot sequences is bounded.http://dx.doi.org/10.1155/S0161171288000407exponential divisorse-perfect numberse-amicable numberse-aliquot sequences.
spellingShingle Peter Hagis
Some results concerning exponential divisors
International Journal of Mathematics and Mathematical Sciences
exponential divisors
e-perfect numbers
e-amicable numbers
e-aliquot sequences.
title Some results concerning exponential divisors
title_full Some results concerning exponential divisors
title_fullStr Some results concerning exponential divisors
title_full_unstemmed Some results concerning exponential divisors
title_short Some results concerning exponential divisors
title_sort some results concerning exponential divisors
topic exponential divisors
e-perfect numbers
e-amicable numbers
e-aliquot sequences.
url http://dx.doi.org/10.1155/S0161171288000407
work_keys_str_mv AT peterhagis someresultsconcerningexponentialdivisors