Variations on Euclid's formula for perfect numbers.
Firoozbakht, Farideh, Hasler, Maximilian F. (2010)
Journal of Integer Sequences [electronic only]
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Firoozbakht, Farideh, Hasler, Maximilian F. (2010)
Journal of Integer Sequences [electronic only]
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Nielsen, Pace P. (2003)
Integers
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McDaniel, Wayne L. (1990)
International Journal of Mathematics and Mathematical Sciences
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Asadulla, Syed (1986)
International Journal of Mathematics and Mathematical Sciences
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Gimbel, Steven, Jaroma, John H. (2003)
Integers
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Zhou, Weiyi, Zhu, Long (2009)
Integers
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Wayne McDaniel (1974)
Acta Arithmetica
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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.
Sándor, József (2004)
JIPAM. Journal of Inequalities in Pure & Applied Mathematics [electronic only]
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McCarthy, Paul J. (1957)
Portugaliae mathematica
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De Koninck, Jean-Marie, Ivić, Aleksandar (1998)
Publications de l'Institut Mathématique. Nouvelle Série
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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.
Davis, Simon (2003)
International Journal of Mathematics and Mathematical Sciences
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