On Deficient-Perfect Numbers

Abstract

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 deficient-perfect number if σ(n) = 2n − d. In this paper, we show that there are no odd deficient-perfect numbers with three distinct prime divisors.2010 Mathematics subject classification: primary 11A25.

On Even Perfect Numbers II

“…From another perspective, Chen [2] defined k-deficient-perfect numbers and determined all odd exactly 2-deficient-perfect numbers with two distinct prime divisors. For more beautiful results on near-perfect numbers and deficient-perfect numbers, see [11,12].…”

On Exactly $3$-Deficient-Perfect Numbers

“…By the definition, n is deficient-perfect if and only if n is exactly 1-deficientperfect. Tang and Feng [33,Lemma 2.1] show that if n is deficient-perfect and n is odd, then n is a square. We can extend their result to the following form.…”

“…In the same year, Tang, Ren, and Li [35] proved that there is no odd nearperfect number n with ω(n) = 3 and found all deficient-perfect numbers m with ω(m) ≤ 2. After that, Tang and Feng [33] extended it by showing that there is no odd deficient-perfect number n with ω(n) = 3. Tang, Ma, and Feng [34] obtained in 2016 the only odd near-perfect number with ω(n) = 4, namely, n = 3 4 • 7 2 • 11 2 • 19 2 , while Sun and He [32] asserted in 2019 that the only odd deficient-perfect number n with ω(n) = 4 is n = 3 2 • 7 2 • 11 2 • 13 2 .…”

On odd deficient-perfect numbers with four distinct prime divisors

“…In 2013, Tang, Ren and Li [11] determined all deficient-perfect numbers with at most two distinct prime factors. In a similar vein, Tang and Feng [9] showed that no odd deficient-perfect number exists with three distinct prime factors. For related problems, see [1-3, 5, 6, 8].…”