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  • MIN TANG (a1) and MIN FENG (a2)


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.


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[1]Anavi, A., Pollack, P. and Pomerance, C., ‘On congruences of the form σ (n) = a (mod n)’, Int. J. Number Theory 9 (2013), 115124.
[2]Cohen, G. L., ‘On odd perfect numbers (II), multiperfect numbers and quasiperfect numbers’, J. Aust. Math. Soc. 29 (1980), 369384.
[3]Hagis, P. and Cohen, G. L., ‘Some results concerning quasiperfect numbers’, J. Aust. Math. Soc. 33 (1982), 275286.
[4]Kishore, M., ‘Odd integers n with five distinct prime factors for which 2 − 10−12 < σ (n)∕n < 2 + 10−12’, Math. Comp. 32 (1978), 303309.
[5]Pollack, P. and Shevelev, V., ‘On perfect and near-perfect numbers’, J. Number Theory 132 (2012), 30373046.
[6]Ren, X. Z. and Chen, Y. G., ‘On near-perfect numbers with two distinct prime factors’, Bull. Aust. Math. Soc. 88 (2013), 520524.
[7]Sándor, J., Mitrinović, D. S. and Crstici, B., Handbook of Number Theory I (Springer, Dordrecht, The Netherlands, 2005).
[8]Tang, M., Ren, X. Z. and Li, M., ‘On near-perfect and deficient-perfect numbers’, Colloq. Math. 133 (2013), 221226.
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