投票: 把全体正整数分成若干不漏不重数字串的方法

王守恩

把全体正整数分成若干不漏不重(不循环)数字串,  譬如:  A+B=全体正整数
A(n)=2, 5, 7, 10, 13, 15, 18, 20, 23, 26, 28, 31, 34, 36, 39, 41, 44, 47, 49, 52, 54, 57, 60, 62, 65, 68,70,
73, 75, 78, 81, 83, 86, 89, 91, 94, 96, 99, 102, 104, 107, 109, 112, 115, 117,120,123,125,128,130,133,
136, 138, 141, 143, 146, 149, 151, 154, 157, 159, 162, 164, 167, 170, 172, 175,178,180,183,185,188,
191, 193, 196, 198, 201, 204, 206, 209, 212, 214, 217, 219, 222, 225, 227, 230, 233, 235, 238, 240,...
\(A(n)=\lfloor\frac{n (3 + \sqrt{5})}{2}\rfloor\)     OEIS--A090909--2023年9月12日
B(n)=1, 3, 4, 6, 8, 9, 11, 12, 14, 16, 17, 19, 21, 22, 24, 25, 27, 29, 30, 32, 33, 35, 37, 38, 40, 42, 43, 45,
46, 48, 50, 51, 53, 55, 56, 58, 59, 61, 63, 64, 66, 67, 69, 71, 72, 74, 76, 77, 79, 80, 82, 84, 85,87,88,90,
92, 93, 95, 97, 98, 100, 101, 103, 105, 106, 108, 110, 111, 113, 114, 116, 118, 119, 121, 122,124,126,
127, 129, 131, \132, 134, 135, 137, 139, 140, 142, 144, 145, 147, 148, 150, 152, 153, 155, 156, ......
\(B(n)=\lfloor\frac{n (1 + \sqrt{5})}{2}\rfloor\)     

譬如:  A+B=全体正整数,   已知A(n)=见下面,   B(n)=？
{A(n)=1, 3, 5, 7, 9, 10, 12, 14, 16, 18, 19, 21, 23, 25, 27, 28, 30, 32, 34, 36, 37, 39, 41, 43, 45,47,48,50,
52, 54, 56, 57, 59, 61, 63, 65, 66, 68, 70, 72, 74, 75, 77, 79, 81, 83, 85, 86, 88, 90, 92, 94,95,97,99,101,
103, 104, 106, 108, 110, 112, 113, 115, 117, 119, 121, 123, 124, 126, 128, 130, 132, 133,135,137,139,
141, 142, 144, 146, 148, 150, 151, 153, 155, 157, 159, 161, 162, 164, 166, 168, 170, 171, 173, 175, ...
\(A(n)=\lfloor\frac{n ( 5+ \sqrt{5})}{4}\rfloor\)     OEIS--108598--2023年10月20日

lihp2020

我记得有个公式
1=1/a+1/b  当a 是无理数时
数列 An=[a*n]   Bn=[b*n]
满足   A+B=全体正整数

lihp2020

刚刚找的一本教材  但是在例题中  
还有一本  是有解题的 我会证明   就不想写了
书名
<初等数论及其应用(原书第5版) / (美)罗森(Rosen， K. H. )著;夏鸿刚译.一北京:机械工业出版社，
2009.6>

lihp2020

1 =1/a+1/b
a=5+sqrt(5)/4
解的b=sqrt(5)
随便算的前几项
2 4 6 8 11 13 15 17 20 22 24 26 29 31 33 35 38 40 42 44 46 49

王守恩

lihp2020 发表于 2023-12-12 06:27
1 =1/a+1/b
a=5+sqrt(5)/4
解的b=sqrt(5)

把全体正整数分成3个不漏不重(不循环)数字串,  譬如:  A+B+C=全体正整数, 好像没有的？

王守恩

{1, 4, 6, 9, 11, 13, 16, 19, 20, 23, 25, 28, 31, 32, 35, 38, 40, 42, 45, 47, 50, 52, 54, 57,59,62,64,66,69,71,
{2, 5, 8, 12, 15, 17, 21, 24, 27, 30, 33, 36, 39, 43, 46, 49, 51, 55, 58, 61, 65, 67, 70, 73, 77, 80,83,86,89,
{3, 7, 10, 14, 18, 22, 26, 29, 34, 37, 41, 44, 48, 53, 56, 60, 63, 68, 72, 75, 79, 82, 87, 90, 94,98,102,106,

1. Table[n + Floor[n*Sqrt[2]/Sqrt[5]] + Floor[n*Sqrt[3]/Sqrt[5]], {n, 1, 33}]
   
2. Table[n + Floor[n*Sqrt[2]/Sqrt[3]] + Floor[n*Sqrt[5]/Sqrt[3]], {n, 1, 33}]
   
3. Table[n + Floor[n*Sqrt[3]/Sqrt[2]] + Floor[n*Sqrt[5]/Sqrt[2]], {n, 1, 33}]

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