有趣的自然数及其性质

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有趣的自然数及其性质之三
数字英文名称中不含字母a的正整数，包括1-999，1000000-1000999，2000000-2000999等等。

A number n is called aban if its name (in English) does not contain the letter "a". The word "and" is not counted and in general I do not use it when I spell out numbers.
如果一个数字n的名称（英文）不包含字母“a”，则该数字   称为aban（音译：阿班数） 。“and”这个词不算在内，一般来说，我在拼出数字时不会使用它。

Among the words used to construct numbers names, only the word "thousand" contains an "a" so the aban numbers are the numbers from 1 to 999, from 1000000 to 1000999, from 2000000 to 2000999, and so on.
在用于构成数字名称的单词中，只有“千”一词包含“a”，因此阿班数是从1到999、从1000000到1000999、从2000000到2000999的数字，依此类推。

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完全数（完美数）
28有因子1,2,4,7,14,28；1+2+4+7+14=28，8是最小的完全数。
12有因子1,2,3,4,6,12；1+2+3+4+6=16>12，12是一个超完全数，又称“盈数”、“丰富数”；
22有因子1,2,11,22；1+2+11=14<22，22是一个欠完全数，又称“亏数”。

完全数(Perfect number)又称完美数或完备数，是一些特殊的自然数，它所有的真因子(即除了自身以外的约数)的和(即因子函数)恰好等于它本身。如果一个数恰好等于它的因子之和，则称该数为"完全数"。

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有趣的自然数及其性质之四
超完全数

abundant numbers  超完全数（丰富数）

A number  $n$  is abundant if  $\sigma(n) > 2\cdot n$, i.e., if the sum of the proper divisors of  $n$  is larger than  $n$.
如果整数n的真因数之和大于n，则该数是丰富的。

For example, 12 is abundant since the sum of its proper divisors, 1 + 2 + 3 + 4 + 6 = 16, exceeds 12 itself.
例如，12 是丰富的，因为它的真因数之和 1 + 2 + 3 + 4 + 6 = 16超过了 12 本身。

There are infinite abundant numbers since, for example, all the multiples of an abundant number is abundant. It has been estimated that about 1/4 of the integers are abundant.
存在无限丰富的数，因为例如，丰富数的所有倍数都是丰富的。据估计，大约有 1/4 的整数是丰富的。

最小的超完全数（丰富数）是 12 , 18 , 20 , 24 , 30 , 36 , 40 , 42 , 48 , 54 , 56 , 60 , 66 , 70 , 72 , 78 , 80 , 84 , 88 , 90 , 96 , 100

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A005101
Abundant numbers (sum of divisors of m exceeds 2m).
数量丰富（m的约数之和超过2m）。——盈数

12, 18, 20, 24, 30, 36, 40, 42, 48, 54, 56, 60, 66, 70, 72, 78, 80, 84, 88, 90, 96, 100, 102, 104, 108, 112, 114, 120, 126, 132, 138, 140, 144, 150, 156, 160, 162, 168, 174, 176, 180, 186, 192, 196, 198, 200, 204, 208, 210, 216, 220, 222, 224, 228, 234, 240, 246, 252, 258, 260, 264, 270

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A005100
Deficient numbers: numbers k such that sigma(k) < 2k.
不足数：数 k 使得 sigma(k) < 2k。——亏数

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 25, 26, 27, 29, 31, 32, 33, 34, 35, 37, 38, 39, 41, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 61, 62, 63, 64, 65, 67, 68, 69, 71, 73, 74, 75, 76, 77, 79, 81, 82, 83, 85, 86

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A000396
Perfect numbers k: k is equal to the sum of the proper divisors of k.
完全数 k：k 等于 k 的真因数之和

1 6
2 28
3 496
4 8128
5 33550336
6 8589869056
7 137438691328
8 2305843008139952128
9 2658455991569831744654692615953842176
10 191561942608236107294793378084303638130997321548169216
11 13164036458569648337239753460458722910223472318386943117783728128
12 14474011154664524427946373126085988481573677491474835889066354349131199152128
13 23562723457267347065789548996709904988477547858392600710143027597506337283178622239730365539602600561360255566462503270175052892578043215543382498428777152427010394496918664028644534128033831439790236838624033171435922356643219703101720713163527487298747400647801939587165936401087419375649057918549492160555646976
14 141053783706712069063207958086063189881486743514715667838838675999954867742652380114104193329037690251561950568709829327164087724366370087116731268159313652487450652439805877296207297446723295166658228846926807786652870188920867879451478364569313922060370695064736073572378695176473055266826253284886383715072974324463835300053138429460296575143368065570759537328128
15 54162526284365847412654465374391316140856490539031695784603920818387206994158534859198999921056719921919057390080263646159280013827605439746262788903057303445505827028395139475207769044924431494861729435113126280837904930462740681717960465867348720992572190569465545299629919823431031092624244463547789635441481391719816441605586788092147886677321398756661624714551726964302217554281784254817319611951659855553573937788923405146222324506715979193757372820860878214322052227584537552897476256179395176624426314480313446935085203657584798247536021172880403783048602873621259313789994900336673941503747224966984028240806042108690077670395259231894666273615212775603535764707952250173858305171028603021234896647851363949928904973292145107505979911456221519899345764984291328

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yangchuanju 发表于 2023-6-29 07:27
A005101
Abundant numbers (sum of divisors of m exceeds 2m).
数量丰富（m的约数之和超过2m）。——盈 ...

除了A005101给出的10000个偶盈数（丰富数）以外，A005231又给出32000个奇盈数
A005231
Odd abundant numbers (odd numbers n whose sum of divisors exceeds 2n).
945, 1575, 2205, 2835, 3465, 4095, 4725, 5355, 5775, 5985, 6435, 6615, 6825, 7245, 7425, 7875, 8085, 8415, 8505, 8925, 9135, 9555, 9765, 10395, 11025, 11655, 12285, 12705, 12915, 13545, 14175, 14805, 15015, 15435, 16065, 16695, 17325, 17955……

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A096399给出10000个相连的盈数对（k和k+1都是盈数）：

A096399
Numbers k such that both k and k+1 are abundant.
5775, 5984, 7424, 11024, 21735, 21944, 26144, 27404, 39375, 43064, 49664, 56924, 58695, 61424, 69615, 70784, 76544, 77175, 79695, 81080, 81675, 82004, 84524, 84644, 89775, 91664, 98175, 103455, 104895, 106784, 109395, 111824, 116655……

最小的连续丰富数对是 ( 5775 , 5776 )。第一个这样的三元组从171078830开始，而由 Bruno Mishutka 发现的已知最小的四元组从 141363708067871564084 949719820472453374 开始。

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A096536
Numbers k such that k, k+1, k+2 are all abundant.
171078830, 268005374, 321893648, 336038624, 487389824, 600350750, 663249950, 668645054, 938109248, 1053424448, 1079741024, 1102433408, 1139364224, 1148927624, 1267293950, 1275861950, 1310259950, 1344330350, 1352253824……

网页全部给出了10^12以内的22583组三元组盈数。

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