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

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.