On the index of an odd perfect number

• Autorzy: Feng-Juan Chen , Yong-Gao Chen
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 11A25: Arithmetic functions; related numbers; inversion formulas
• 11B83: Special sequences and polynomials

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Colloquium Mathematicum
• Rocznik: 2014
• Tom: 136
• Numer: 1
• Strony: 41-49

Daty

wydano

2014

Twórcy

autor

Feng-Juan Chen

• School of Mathematical Sciences, Soochow University, Suzhou 215006, China

autor

Yong-Gao Chen

• School of Mathematical Sciences, and Institute of Mathematics, Nanjing Normal University, Nanjing 210023, China

Identyfikatory

DOI

10.4064/cm136-1-4