On quasi-imperfect numbers with at most four distinct prime divisors

Cui-Fang Sun (), Zhao-Cheng He () and Tian-Tian Tao ()
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Cui-Fang Sun: Anhui Normal University
Zhao-Cheng He: Anhui Normal University
Tian-Tian Tao: Anhui Normal University

Indian Journal of Pure and Applied Mathematics, 2021, vol. 52, issue 2, 429-438

Abstract: Abstract Let $$\rho$$ ρ be a multiplicative arithmetic function defined by $$\rho (p^{\alpha })=p^{\alpha }-p^{\alpha -1}+p^{\alpha -2}-\cdots +(-1)^{\alpha }$$ ρ ( p α ) = p α - p α - 1 + p α - 2 - ⋯ + ( - 1 ) α for a prime power $$p^{\alpha }$$ p α . For a positive integer n, we call n a quasi-imperfect number if $$2\rho (n)=n+1$$ 2 ρ ( n ) = n + 1 . In this paper, we show that there are only four quasi-imperfect numbers with at most four distinct prime divisors. We also pose some conjectures for further research.

Keywords: multiplicative arithmetic function; imperfect number; quasi-imperfect number; Primary; 11A25 (search for similar items in EconPapers)
Date: 2021
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DOI: 10.1007/s13226-021-00048-1

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