殆完全数

殆完全数（almost perfect number）是一种特别的自然数，它所有的真约数（即除了自身以外的约数）的和，恰好等于它本身减一。若用除数函数（其真约数的和及其本身）来表示，若一自然数n的除数函数σ(n)等于2n - 1，该自然数即为殆完全数。殆完全数是一种亏数。亏度（σ(n) − 2n）为-1。

例如4的除数函数为2+1=3，比4小1，因此4是殆完全数。

目前已知的殆完全数为2的非负次幂（OEIS数列A000079），因此唯一已知奇数的殆完全数为2⁰ = 1，但尚未证明除了2的非负次幂以外，是否存在其他型式的殆完全数。可以证明若存在大于1的奇数殆完全数，至少会有六个素因数^[1]^[2]。

若m是奇数殆完全数，则m(2m − 1)会是笛卡尔数^[3]，而且，若a和b满足 $b+3<a<{\sqrt {m/2}}$ ，且4m − a and 4m + b都是素数，则m(4m − a)(4m + b)会是奇数的奇异数^[4]。

参见

参考资料

Richard K. Guy|Guy, R. K., Almost Perfect, Quasi-Perfect, Pseudoperfect, Harmonic, Weird, Multiperfect and Hyperperfect Numbers. §B2 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 16 and 45-53, 1994.
Singh, S., Fermat's Enigma: The Epic Quest to Solve the World's Greatest Mathematical Problem. New York: Walker, p. 13, 1997.

^ Kishore, Masao. Odd integers N with five distinct prime factors for which 2−10⁻¹² < σ(N)/N < 2+10⁻¹² (PDF). Mathematics of Computation. 1978, 32: 303–309. ISSN 0025-5718. JSTOR 2006281. MR 0485658. Zbl 0376.10005. doi:10.2307/2006281.
^ Kishore, Masao. On odd perfect, quasiperfect, and odd almost perfect numbers. Mathematics of Computation. 1981, 36 (154): 583–586. ISSN 0025-5718. JSTOR 2007662. Zbl 0472.10007. doi:10.2307/2007662 .
^ Banks, William D.; Güloğlu, Ahmet M.; Nevans, C. Wesley; Saidak, Filip. Descartes numbers. De Koninck, Jean-Marie; Granville, Andrew; Luca, Florian (编). Anatomy of integers. Based on the CRM workshop, Montreal, Canada, March 13–17, 2006. CRM Proceedings and Lecture Notes 46. Providence, RI: American Mathematical Society. 2008: 167–173. ISBN 978-0-8218-4406-9. Zbl 1186.11004.
^ Melfi, Giuseppe. On the conditional infiniteness of primitive weird numbers. Journal of Number Theory. 2015, 147: 508–514. doi:10.1016/j.jnt.2014.07.024 .

外部链接

埃里克·韦斯坦因. Almost perfect number. MathWorld.

[Kis1978-1] Kishore, Masao. Odd integers N with five distinct prime factors for which 2−10⁻¹² < σ(N)/N < 2+10⁻¹² (PDF). Mathematics of Computation. 1978, 32: 303–309. ISSN 0025-5718. JSTOR 2006281. MR 0485658. Zbl 0376.10005. doi:10.2307/2006281.

[Kis1981-2] Kishore, Masao. On odd perfect, quasiperfect, and odd almost perfect numbers. Mathematics of Computation. 1981, 36 (154): 583–586. ISSN 0025-5718. JSTOR 2007662. Zbl 0472.10007. doi:10.2307/2007662 .

[BGNS-3] Banks, William D.; Güloğlu, Ahmet M.; Nevans, C. Wesley; Saidak, Filip. Descartes numbers. De Koninck, Jean-Marie; Granville, Andrew; Luca, Florian (编). Anatomy of integers. Based on the CRM workshop, Montreal, Canada, March 13–17, 2006. CRM Proceedings and Lecture Notes 46. Providence, RI: American Mathematical Society. 2008: 167–173. ISBN 978-0-8218-4406-9. Zbl 1186.11004.

[4] Melfi, Giuseppe. On the conditional infiniteness of primitive weird numbers. Journal of Number Theory. 2015, 147: 508–514. doi:10.1016/j.jnt.2014.07.024 .

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查论编和约数有关的整数分类
简介	素因数分解约数元约数除数函数素因数算术基本定理
依约数分解分类	素数合数半素数普洛尼克数楔形数无平方数因数的数幂数素数幂平方数立方数次方数阿喀琉斯数光滑数正规数粗糙数不寻常数
依约数和分类	完全数殆完全数准完全数多重完全数 Hemiperfect数 Hyperperfect number（英语：Hyperperfect number）超完全数元完全数半完全数本原半完全数实际数
有许多约数	过剩数本原过剩数高过剩数超过剩数可罗萨里过剩数高合成数 Superior highly composite number（英语：Superior highly composite number）奇异数
和真因子和数列有关	不可及数相亲数交际数婚约数
其他	亏数友谊数孤独数卓越数欧尔调和数佩服数节俭数等数位数奢侈数