ON DEFICIENT-PERFECT NUMBERS. Issue 2 (October 2014)

• Record Type: Journal Article
• Authors: TANG, MIN; FENG, MIN
• Is Part Of: Bulletin of the Australian Mathematical Society. Volume 90:Issue 2(2014)
• Page Start: 186
• Page End: 194
• Publication Date: 2014-10
• Subjects: Mathematics -- Societies, etc; Mathematics -- Periodicals; 510.5
• DOI: 10.1017/S0004972714000082
• Languages: English
• ISSNs: 0004-9727

Abstract

<abstract> <title>Abstract</title> <p>For a positive integer <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgh1308vg0p" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}n$]]></tex-math></alternatives></inline-formula>, let <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgh1308vjsd" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$\sigma (n)$]]></tex-math></alternatives></inline-formula> denote the sum of the positive divisors of <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgh1308vhwj" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$n$]]></tex-math></alternatives></inline-formula>. Let <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgh1308vjb7" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$d$]]></tex-math></alternatives></inline-formula> be a proper divisor of <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgh1308vh95" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$n$]]></tex-math></alternatives></inline-formula>. We call <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgh1308vkdv" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$n$]]></tex-math></alternatives></inline-formula> a deficient-perfect number if <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgh1308vjtz" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$\sigma (n) = 2n - d$]]></tex-math></alternatives></inline-formula>. In this paper, we show that there are no odd deficient-perfect numbers with three distinct prime divisors.</p> </abstract>