Prime numbers such that the sums of the divisors of their powers are perfect squares
A. Takaku (1984)
Colloquium Mathematicae
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A. Takaku (1984)
Colloquium Mathematicae
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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.
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.
Asadulla, Syed (1986)
International Journal of Mathematics and Mathematical Sciences
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McDaniel, Wayne L. (1990)
International Journal of Mathematics and Mathematical Sciences
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Sándor, József (2004)
JIPAM. Journal of Inequalities in Pure & Applied Mathematics [electronic only]
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Gimbel, Steven, Jaroma, John H. (2003)
Integers
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Lajos Hajdu, Shanta Laishram, Szabolcs Tengely (2016)
Acta Arithmetica
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We investigate power values of sums of products of consecutive integers. We give general finiteness results, and also give all solutions when the number of terms in the sum considered is at most ten.
Cohen, Graeme L., te Riele, Herman J.J. (1996)
Experimental Mathematics
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Sándor, József, Kovács, Lehel István (2009)
Acta Universitatis Sapientiae. Mathematica
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Nielsen, Pace P. (2003)
Integers
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De Koninck, Jean-Marie, Ivić, Aleksandar (1998)
Publications de l'Institut Mathématique. Nouvelle Série
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Le Anh Vinh, Dang Phuong Dung (2006)
JIPAM. Journal of Inequalities in Pure & Applied Mathematics [electronic only]
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Tomohiro Yamada (2005)
Colloquium Mathematicae
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We show that there is an effectively computable upper bound of odd perfect numbers whose Euler factors are powers of fixed exponent.
Hagis, Peter jun. (1988)
International Journal of Mathematics and Mathematical Sciences
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