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ON NEAR-PERFECT NUMBERS OF SPECIAL FORMS
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 108 / Issue 3 / December 2023
- Published online by Cambridge University Press:
- 14 February 2023, pp. 366-372
- Print publication:
- December 2023
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ON DEFICIENT-PERFECT NUMBERS
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 90 / Issue 2 / October 2014
- Published online by Cambridge University Press:
- 23 May 2014, pp. 186-194
- Print publication:
- October 2014
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For a positive integer
$\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}n$, let 
$\sigma (n)$ denote the sum of the positive divisors of 
$n$. Let 
$d$ be a proper divisor of 
$n$. We call 
$n$ a deficient-perfect number if 
$\sigma (n) = 2n - d$. In this paper, we show that there are no odd deficient-perfect numbers with three distinct prime divisors.
ON NEAR-PERFECT NUMBERS WITH TWO DISTINCT PRIME FACTORS
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 88 / Issue 3 / December 2013
- Published online by Cambridge University Press:
- 11 March 2013, pp. 520-524
- Print publication:
- December 2013
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Recently, Pollack and Shevelev [‘On perfect and near-perfect numbers’, J. Number Theory 132 (2012), 3037–3046] introduced the concept of near-perfect numbers. A positive integer
$n$ is called near-perfect if it is the sum of all but one of its proper divisors. In this paper, we determine all near-perfect numbers with two distinct prime factors.
