Abstract
For any positive integer n, let \(\sigma (\mathrm{n})\) and p(n) denote the sum of divisors and the least prime divisor of n respectively. Let a, b be positive integers. In this paper we prove the following two results: (i) If 4 | a and \(\gcd (a, b)=1\), then a and b do not satisfy \(\sigma (a)= \sigma (b)=a+b\). (ii) If \(a>10^{8}\) and \(p(a)>2\log _{2}a+1\), where \(\log _{2}{a}\) is the logarithm of a with base 2, then a and b do not satisfy \(\sigma (a)=\sigma (b)=a+b+\lambda \), where \(\lambda \in \{0,\pm 1\}\).
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Acknowledgments
The author would like to thank the referee for his very helpful and detailed comments which have significantly improved the presentation of this paper. This work is supported by N.S.F.(61373117) and P.N.S.F.(2014JM1009) of P.R.China.
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Xiaoyan, G., Guohua, G. & deli, L. A note on amicable numbers and their variations. Period Math Hung 72, 180–185 (2016). https://doi.org/10.1007/s10998-016-0118-3
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DOI: https://doi.org/10.1007/s10998-016-0118-3