On the index of an odd perfect number

Feng-Juan Chen, and Yong-Gao Chen. "On the index of an odd perfect number." Colloquium Mathematicae 136.1 (2014): 41-49. <http://eudml.org/doc/284362>.

@article{Feng2014,
abstract = {Suppose that N is an odd perfect number and $q^\{α\}$ is a prime power with $q^\{α\} || N$. Define the index $m = σ(N/q^\{α\})/q^\{α\}$. We prove that m cannot take the form $p^\{2u\}$, where u is a positive integer and 2u+1 is composite. We also prove that, if q is the Euler prime, then m cannot take any of the 30 forms q₁, q₁², q₁³, q₁⁴, q₁⁵, q₁⁶, q₁⁷, q₁⁸, q₁q₂, q₁²q₂, q₁³q₂, q₁⁴ q₂, q₁⁵q₂, q₁²q₂², q₁³q₂², q₁⁴q₂², q₁q₂q₃, q₁²q₂q₃, q₁³q₂q₃, q₁⁴q₂q₃, q₁²q₂²q₃, q₁²q₂²q₃², q₁q₂q₃q₄, q₁²q₂q₃q₄, q₁³q₂q₃q₄, q₁²q₂²q₃q₄, q₁q₂q₃q₄q₅, q₁²q₂q₃q₄q₅, q₁q₂q₃q₄q₅q₆, q₁q₂q₃q₄q₅q₆q₇, where q₁, q₂, q₃, q₄, q₅, q₆, q₇ are distinct odd primes. A similar result is proved if q is not the Euler prime. These extend recent results of Broughan, Delbourgo, and Zhou. We also pose a related problem.},
author = {Feng-Juan Chen, Yong-Gao Chen},
journal = {Colloquium Mathematicae},
keywords = {odd perfect number; Euler prime},
language = {eng},
number = {1},
pages = {41-49},
title = {On the index of an odd perfect number},
url = {http://eudml.org/doc/284362},
volume = {136},
year = {2014},
}

TY - JOUR
AU - Feng-Juan Chen
AU - Yong-Gao Chen
TI - On the index of an odd perfect number
JO - Colloquium Mathematicae
PY - 2014
VL - 136
IS - 1
SP - 41
EP - 49
AB - Suppose that N is an odd perfect number and $q^{α}$ is a prime power with $q^{α} || N$. Define the index $m = σ(N/q^{α})/q^{α}$. We prove that m cannot take the form $p^{2u}$, where u is a positive integer and 2u+1 is composite. We also prove that, if q is the Euler prime, then m cannot take any of the 30 forms q₁, q₁², q₁³, q₁⁴, q₁⁵, q₁⁶, q₁⁷, q₁⁸, q₁q₂, q₁²q₂, q₁³q₂, q₁⁴ q₂, q₁⁵q₂, q₁²q₂², q₁³q₂², q₁⁴q₂², q₁q₂q₃, q₁²q₂q₃, q₁³q₂q₃, q₁⁴q₂q₃, q₁²q₂²q₃, q₁²q₂²q₃², q₁q₂q₃q₄, q₁²q₂q₃q₄, q₁³q₂q₃q₄, q₁²q₂²q₃q₄, q₁q₂q₃q₄q₅, q₁²q₂q₃q₄q₅, q₁q₂q₃q₄q₅q₆, q₁q₂q₃q₄q₅q₆q₇, where q₁, q₂, q₃, q₄, q₅, q₆, q₇ are distinct odd primes. A similar result is proved if q is not the Euler prime. These extend recent results of Broughan, Delbourgo, and Zhou. We also pose a related problem.
LA - eng
KW - odd perfect number; Euler prime
UR - http://eudml.org/doc/284362
ER -