New amicable pairs of type $(2,2)$ and type $(3,2)$

Costello, Patrick

doi:10.1090/s0025-5718-02-01414-x

Search citation statements

Order By: Relevance

Paper Sections

Select...

Introduction1

Citation Types

Supporting

Mentioning

Contrasting

Year Published

2004

2016

Publication Types

Select...

Article1

Book1

Relationship

Self Cite0

Independent2

Authors

Journals

Cited by 2 publications

(1 citation statement)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It is not known if a pair of amicable numbers (a, b) exists with a and b of opposite parity, or with gcd(a, b) = 1. In this respect, we prove the following result: For any positive integer n with n > 1, let p(n) denote the least prime divisor of n. We observe from the known pairs (a, b) of amicable numbers and their variations that p(a) and p(b) are small compared with the sizes of a and b, respectively (see [1][2][3][4][5][6][7]11,12,15] and [16]). On the other hand, it has been known that, for some special positive integers a, if p(a) is large enough, then a must be an anti-sociable number (see [9,10] and [14]).…”

Section: Introductionmentioning

confidence: 99%

A note on amicable numbers and their variations

Guo¹,

Geng²,

Lei

2016

Period Math Hung

View full text Add to dashboard Cite

For any positive integer n, let σ (n) and p(n) denote the sum of divisors and the least prime divisor of n respectively. Let a, b be positive integers. In this paper we prove the following two results: (i) If 4|a and gcd(a, b) = 1, then a and b do not satisfy σ (a) = σ (b) = a + b. (ii) If a > 10 8 and p(a) > 2 log 2 a + 1, where log 2 a is the logarithm of a with base 2, then a and b do not satisfy σ (a) = σ (b) = a + b + λ, where λ ∈ {0, ±1}.

show abstract