Displaying similar documents to “Abundancy “outlaws” of the form ( σ ( N ) + t ) N .”

On near-perfect numbers

Min Tang, Xiaoyan Ma, Min Feng (2016)

Colloquium Mathematicae

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For a positive integer n, let σ(n) denote the sum of the positive divisors of n. We call n a near-perfect number if σ(n) = 2n + d where d is a proper divisor of n. We show that the only odd near-perfect number with four distinct prime divisors is 3⁴·7²·11²·19².

On near-perfect and deficient-perfect numbers

Min Tang, Xiao-Zhi Ren, Meng Li (2013)

Colloquium Mathematicae

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For a positive integer n, let σ(n) denote the sum of the positive divisors of n. Let d be a proper divisor of n. We call n a near-perfect number if σ(n) = 2n + d, and a deficient-perfect number if σ(n) = 2n - d. We show that there is no odd near-perfect number with three distinct prime divisors and determine all deficient-perfect numbers with at most two distinct prime factors.

On a sum of divisors problem.

De Koninck, Jean-Marie, Ivić, Aleksandar (1998)

Publications de l'Institut Mathématique. Nouvelle Série

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Odd perfect polynomials over 𝔽 2

Luis H. Gallardo, Olivier Rahavandrainy (2007)

Journal de Théorie des Nombres de Bordeaux

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A perfect polynomial over 𝔽 2 is a polynomial A ∈ 𝔽 2 [ x ] that equals the sum of all its divisors. If gcd ( A , x 2 + x ) = 1 then we say that A is odd. In this paper we show the non-existence of odd perfect polynomials with either three prime divisors or with at most nine prime divisors provided that all exponents are equal to 2 .