﻿ 具有四个素因子的奇亏完全数

具有四个素因子的奇亏完全数On Odd Deficient-Perfect Numbers with Four Distinct Prime Divisors

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 . On the basis of the references, we characterize some properties of odd deficient-perfect numbers with four distinct prime divisors. We prove that if is an odd deficient-perfect number, then p1 = 3, p2 ≤ 13, and improve the result of the references.

[1] Guy, R.K. (1981) Unsolved Problems in Number Theory. Springer-Verlag, New York, 25-56. http://dx.doi.org/10.1007/978-1-4757-1738-9

[2] 华罗庚. 数论导引[M]. 北京: 科学出版社, 1979: 13-14.

[3] Erdos, P. (1959) Remarks on Number Theory II: Some Problems on the σ Function. Acta Arithmetica, 5, 171-177.

[4] Poliack, P. and Shevelev, P. (2012) On Perfect and Near-Perfect Numbers. Journal of Number Theory, 132, 3037-3046. http://dx.doi.org/10.1016/j.jnt.2012.06.008

[5] Abbott, H.L., Aull, C.E., Brown, E., et al. (1973) Quasi Perfect Numbers. Acta Arithmetica, 22, 439-447.

[6] Cattaneo, P. (1951) Sui Numeriquasi Perfetti. Boll.Un. Mat. Ital., 6, 59-62.

[7] Pomerance, C. (1975) On the Congrunces σ(n) ≡ a (mod n) and n ≡ a (mod ϕ(n)). Acta Arithmetica, 26, 265-272.

[8] Kishore, M. (1978) Odd Integers n with Five Distinct Prime Factors for Which 2-10−12<σ(n)/n< 2+10−12. Mathematics of Computation, 32, 303-309. http://dx.doi.org/10.2307/2006281

[9] Cohen, G.L. (1980) On Odd Perfect Numbers. II. Multiperfect Numbers and Quasiperfect Numbers. Journal of the Australian Mathematical Society Series A, 29, 369-384. http://dx.doi.org/10.1017/S1446788700021376

[10] Kishore, M. (1981) On Odd Perfect, Quasiperfect, and Odd Almost Perfect Numbers. Mathematics of Computation, 36, 583-586. http://dx.doi.org/10.1090/S0025-5718-1981-0606516-3

[11] Cohen, G.L. (1982) The Nonexistence of Quasiperfect Numbers of Certain Forms. The Fibonacci Quarterly, 20, 81-84.

[12] Hagis, P. and Cohen, G.L. (1982) Some Results Concerning Quasiperfect Numbers. Journal of the Australian Mathematical Society Series A, 33, 275-286. http://dx.doi.org/10.1017/S1446788700018401

[13] Tang, M. and Li, M. (2012) On the Congruence σ(n) ≡ 1 (mod n). Journal of Mathematical Research with Applications, 32, 673-676.

[14] Anavi, A., Pollack, P. and Pomerance, C. (2013) On Congrunces of the Form σ(n) ≡ a (mod n). International Journal of Number Theory, 9, 115-124. http://dx.doi.org/10.1142/S1793042112501266

[15] Li, M. and Tang, M. (2014) On the Congruence σ(n) ≡ 1 (mod n) II. Journal of Mathematical Research with Applications, 34, 155-160.

[16] Tang, M. and Feng, M. (2014) On Deficient-Perfect Numbers. Bulletin of the Australian Mathematical Society, 90, 186-194. http://dx.doi.org/10.1017/S0004972714000082

[17] Ren, X.Z. and Chen, Y.G. (2013) On Near-Perfect Numbers with Two Distinct Prime Factors. Bulletin of the Australian Mathematical Society, 88, 520-524. http://dx.doi.org/10.1017/S0004972713000178

[18] Tang, M., Ren, X.Z. and Li, M. (2013) On Near-Perfect and Deficient-Perfect Numbers. Colloquium Mathematicum, 133, 221-226. http://dx.doi.org/10.4064/cm133-2-8

[19] 马小艳, 王玉杰. 具有四个素因子的奇亏完全数[J]. 纯粹数学与应用数学, 2015, 31(6): 643-649.

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