More on the total number of prime factors of an odd perfect number

Hare, Kathryn E.

doi:10.1090/s0025-5718-04-01683-7

Cited by 10 publications

(21 citation statements)

References 7 publications

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“…Iannucci and Sorli showed that Ω(N ) ≥ 37 [8]. The author extended this to give Ω(N ) ≥ 47 [7]. This paper extends this result to give…”

Section: Introductionmentioning

confidence: 56%

“…There are two main modifications to the algorithm in [7]. First, as opposed to doing every individual case of [α,…”

Section: Then We Havementioning

confidence: 99%

“…It should be pointed out that this modification introduced a new means of obtaining a contradiction, which in [7] was taken care of in the choice of partitions. This is listed as contradiction (5) below.…”

Section: Then We Havementioning

confidence: 99%

“…In [7] there was a specific roadblock which prevented any further calculation. This is no longer the case.…”

Section: Introductionmentioning

confidence: 99%

“…As was done in [7], numbers were factored using the ifactor command in MAPLE, with the easy option specified. If easy factors were not found, then the number was checked in a hints database (which currently contains over 700 completely, or partially factored numbers).…”

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confidence: 99%

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New techniques for bounds on the total number of prime factors of an odd perfect number

Hare¹

2007

Math. Comp.

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“…Iannucci and Sorli showed that Ω(N ) ≥ 37 [8]. The author extended this to give Ω(N ) ≥ 47 [7]. This paper extends this result to give…”

Section: Introductionmentioning

confidence: 56%

“…There are two main modifications to the algorithm in [7]. First, as opposed to doing every individual case of [α,…”

Section: Then We Havementioning

confidence: 99%

Section: Then We Havementioning

confidence: 99%

“…In [7] there was a specific roadblock which prevented any further calculation. This is no longer the case.…”

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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New techniques for bounds on the total number of prime factors of an odd perfect number

Hare¹

2007

Math. Comp.

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The First Years

Narkiewicz

2012

Springer Monographs in Mathematics

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On the classification of integers n that divide ϕ(n)+σ(n)

Harris

2009

Journal of Number Theory

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The expressions φ(n) + σ (n) − 3n and φ(n) + σ (n) − 4n are unusual among linear combinations of arithmetic functions in that they each vanish on a nonempty set of composite numbers. In 1966, Nicol proved that the set A := {n | (φ(n) + σ (n))/n ∈ N 3 } contains 2 a · 3 · (2 a−2 · 7 − 1) if and only if 2 a−2 · 7 − 1 is prime and conjectured that A contains no odd integers. A 2008 paper by Luca and Sandor completely classifies the elements of A that have three distinct prime factors and observes that Nicol's conjecture holds for numbers with fewer than six distinct prime factors. In this paper we let A K denote the set of n ∈ A with exactly K distinct prime factors and present a computer-implementable algorithm that decides whether Nicol's conjecture holds for a given A K . Using this algorithm, we verify Nicol's conjecture for A 6 and completely classify the elements of A 4 . We prove that all but finitely many n ∈ A 4 have the form 2 a · 3 · p 3 · p 4 , and that all but finitely many n ∈ A 5 are divisible by 6 and not 9. In addition, we prove that every A K is contained in a finite union of sequences that each have the form {pp 1 , . . . , p k are distinct primes, and each a ji as well as the least prime factor of w i go to infinity as i does.

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More on the total number of prime factors of an odd perfect number

Abstract: Abstract. Let σ(n) denote the sum of the positive divisors of n. We say that n is perfect if σ(n) = 2n. Currently there are no known odd perfect numbers. It is known that if an odd perfect number exists, then it must be of the form N = p α k j=1 q

Cited by 10 publications

References 7 publications

New techniques for bounds on the total number of prime factors of an odd perfect number

New techniques for bounds on the total number of prime factors of an odd perfect number

The First Years

On the classification of integers n that divide ϕ(n)+σ(n)

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