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在不同的整数分解树人们可能会想到递归平方分解到素数的二叉树(递归分解为中心因子直至素数的二叉树)看起来最有趣。
递归平方分解到素数的二叉树
这个平方分解,即进入核心因素第,页,共页最小化和。它对应于面积为。这有最小的半周长(A063655美元),自在以下情况下最小。48/ \6 8/ \ / \2 3 2 4/ \2 2个
这个二叉树递归平方因子分解,即成为中心因子,直至素数
- 素数作为叶节点;
- 奇数节点(除根节点外,所有节点都是成对的中心因子);
- 偶数边缘(显然,所有边都是成对的)。
递归平方分解二叉树的节点数是节点(n个)=1+总和(节点(我),个节点(j个)). |
递归平方分解二叉树的层数是水平(n个)=1+最大值(水平(我),个级别(j个)). |
A162348号n的中心因子对(i,j),例如i*j=n,其中i是n的最大因子,j是n的最小因子。
- {1, 1, 1, 2, 1, 3, 2, 2, 1, 5, 2, 3, 1, 7, 2, 4, 3, 3, 2, 5, 1, 11, 3, 4, 1, 13, 2, 7, 3, 5, 4, 4, 1, 17, 3, 6, 1, 19, 4, 5, 3, 7, 2, 11, 1, 23, 4, 6, 5, 5, 2, 13, 3, 9, 4, 7, 1, 29, 5, 6, 1, 31, 4, ...}
A033676号n的最大除数<=sqrt(n)。
- {1, 1, 1, 2, 1, 2, 1, 2, 3, 2, 1, 3, 1, 2, 3, 4, 1, 3, 1, 4, 3, 2, 1, 4, 5, 2, 3, 4, 1, 5, 1, 4, 3, 2, 5, 6, 1, 2, 3, 5, 1, 6, 1, 4, 5, 2, 1, 6, 7, 5, 3, 4, 1, 6, 5, 7, 3, 2, ...}
A033677号n>=sqrt(n)的最小除数。
- {1,2,3,2,5,3,7,4,3,5,11,4,13,7,5,4,17,6,19,5,7,11,23,6,5,13,9,7,29,6,31,8,11,17,7,6,37,19,13,8,41,7,43,11,9,23,47,8,7,10,17,…}
递归平方分解的二叉树
|
|
节点 |
边缘 (节点-1) |
水平 |
高度 (级别-1) |
1
|
清空产品
|
0 |
|
0 |
|
2
|
2 |
1 |
0 |
1 |
0 |
三
|
三 |
1 |
0 |
1 |
0 |
4
|
4 → 2 * 2 |
三 |
2 |
2 |
1 |
5
|
5 |
1 |
0 |
1 |
0 |
6
|
6 → 2 * 3 |
三 |
2 |
2 |
1 |
7
|
7 |
1 |
0 |
1 |
0 |
8
|
8 → 2 * 4 → 2 * (2 * 2) |
5 |
4 |
三 |
2 |
9
|
9 → 3 * 3 |
三 |
2 |
2 |
1 |
10
|
10 → 2 * 5 |
三 |
2 |
2 |
1 |
11
|
11 |
1 |
0 |
1 |
0 |
12
|
12 → 3 * 4 → 3 * (2 * 2) |
5 |
4 |
三 |
2 |
13
|
13 |
1 |
0 |
1 |
0 |
14
|
14 → 2 * 7 |
三 |
2 |
2 |
1 |
15
|
15 → 3 * 5 |
三 |
2 |
2 |
1 |
16
|
16 → 4 * 4 → (2 * 2) * (2 * 2) |
7 |
6 |
三 |
2 |
17
|
17 |
1 |
0 |
1 |
0 |
18
|
18 → 3 * 6 → 3 * (2 * 3) |
5 |
4 |
三 |
2 |
19
|
19 |
1 |
0 |
1 |
0 |
20
|
20 → 4 * 5 → (2 * 2) * 5 |
5 |
4 |
三 |
2 |
21
|
21 → 3 * 7 |
三 |
2 |
2 |
1 |
22
|
22 → 2 * 11 |
三 |
2 |
2 |
1 |
23
|
23 |
1 |
0 |
1 |
0 |
24
|
24 → 4 * 6 → (2 * 2) * (2 * 3) |
7 |
6 |
三 |
2 |
25
|
25 → 5 * 5 |
三 |
2 |
2 |
1 |
26
|
26 → 2 * 13 |
三 |
2 |
2 |
1 |
27
|
27 → 3 * 9 → 3 * (3 * 3) |
5 |
4 |
三 |
2 |
28
|
28 → 4 * 7 → (2 * 2) * 7 |
5 |
4 |
三 |
2 |
29
|
29 |
1 |
0 |
1 |
0 |
30
|
30 → 5 * 6 → 5 * (2 * 3) |
5 |
4 |
三 |
2 |
|
|
|
节点 |
边缘 (节点-1) |
水平 |
高度 (级别-1) |
31
|
31 |
1 |
0 |
1 |
0 |
32
|
32 → 4 * 8 → (2 * 2) * (2 * 4) → (2 * 2) * (2 * (2 * 2)) |
9 |
8 |
4 |
三 |
33
|
33 → 3 * 11 |
三 |
2 |
2 |
1 |
34
|
34→2*17 |
三 |
2 |
2 |
1 |
35
|
35 → 5 * 7 |
三 |
2 |
2 |
1 |
36
|
36 → 6 * 6 → (2 * 3) * (2 * 3) |
7 |
6 |
三 |
2 |
37
|
37 |
1 |
0 |
1 |
0 |
38
|
38 → 2 * 19 |
三 |
2 |
2 |
1 |
39
|
39 → 3 * 13 |
三 |
2 |
2 |
1 |
40
|
40→5*8→5*(2*4)→5*(2*(2*2)) |
7 |
6 |
4 |
三 |
41
|
41 |
1 |
0 |
1 |
0 |
42
|
42 → 6 * 7 → (2 * 3) * 7 |
5 |
4 |
三 |
2 |
43
|
43 |
1 |
0 |
1 |
0 |
44
|
44 → 4 * 11 → (2 * 2) * 11 |
5 |
4 |
三 |
2 |
45
|
45 → 5 * 9 → 5 * (3 * 3) |
5 |
4 |
三 |
2 |
46
|
46 → 2 * 23 |
三 |
2 |
2 |
1 |
47
|
47 |
1 |
0 |
1 |
0 |
48
|
48 → 6 * 8 → (2 * 3) * (2 * 4) → (2 * 3) * (2 * (2 * 2)) |
9 |
8 |
4 |
三 |
49
|
49 → 7 * 7 |
三 |
2 |
2 |
1 |
50
|
50 → 5 * 10 → 5 * (2 * 5) |
5 |
4 |
三 |
2 |
51
|
51 → 3 * 17 |
三 |
2 |
2 |
1 |
52
|
52 → 4 * 13 → (2 * 2) * 13 |
5 |
4 |
三 |
2 |
53
|
53 |
1 |
0 |
1 |
0 |
54
|
54 → 6 * 9 → (2 * 3) * (3 * 3) |
7 |
6 |
三 |
2 |
55
|
55 → 5 * 11 |
三 |
2 |
2 |
1 |
56
|
56 → 7 * 8 → 7 * (2 * 4) → 7 * (2 * (2 * 2)) |
7 |
6 |
4 |
三 |
57
|
57 → 3 * 19 |
三 |
2 |
2 |
1 |
58
|
58 → 2 * 29 |
三 |
2 |
2 |
1 |
59
|
59 |
1 |
0 |
1 |
0 |
60
|
60 → 6 * 10 → (2 * 3) * (2 * 5) |
7 |
6 |
三 |
2 |
|
递归平方分解的二叉树具有k个边缘对
|
顺序 |
A-数字
|
0
|
底漆: {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, ...} |
A000040型
|
1
|
双三聚体: {4, 6, 9, 10, 14, 15, 21, 22, 25, 26, 33, 34, 35, 38, 39, 46, 49, 51, 55, 57, 58, 62, 65, 69, 74, 77, 82, 85, 86, 87, 91, 93, 94, 95, 106, 111, 115, ...} |
A001358号
|
2
|
{8, 12, 18, 20, 27, 28, 30, 42, 44, 45, 50, 52, ...} |
A?????? |
三
|
{16, 24, 36, 40, 54, 60, ...} |
A?????? |
4
|
{32, 48, ...} |
A?????? |
5
|
{?, ...} |
A?????? |
6
|
{?, ...} |
A?????? |
7
|
{?, ...} |
A?????? |
8
|
{?, ...} |
A?????? |
9
|
{?, ...} |
A?????? |
10
|
{?, ...} |
A?????? |
11
|
{?, ...} |
A?????? |
12
|
{?, ...} |
A?????? |
13
|
{?, ...} |
A?????? |
14
|
{?, ...} |
A?????? |
15
|
{?, ...} |
A?????? |
16
|
{?, ...} |
A?????? |
17
|
{?, ...} |
A?????? |
18
|
{?, ...} |
A?????? |
19
|
{?, ...} |
A?????? |
20
|
{?, ...} |
A?????? |
21
|
{?, ...} |
A?????? |
22
|
{?, ...} |
A?????? |
23
|
{?, ...} |
A?????? |
24
|
{?, ...} |
A?????? |
25
|
{?, ...} |
A?????? |
26
|
{?, ...} |
A?????? |
27
|
{?, ...} |
A?????? |
28
|
{?, ...} |
A?????? |
29
|
{?, ...} |
A?????? |
30
|
{?, ...} |
A?????? |
递归平方分解的二叉树具有k个水平(因此为高度k个− 1)
|
顺序 |
A-数字
|
0
|
{1} |
|
1
|
底漆: {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, ...} |
A000040型
|
2
|
双三聚体: {4, 6, 9, 10, 14, 15, 21, 22, 25, 26, 33, 34, 35, 38, 39, 46, 49, 51, 55, 57, 58, 62, 65, 69, 74, 77, 82, 85, 86, 87, 91, 93, 94, 95, 106, 111, 115, ...} |
A001358号
|
三
|
{8, 12, 16, 18, 20, 24, 27, 28, 30, 36, 42, 44, 45, 50, 52, 54, 56, 60, ...} |
A?????? |
4
|
{32、40、48、56、…} |
A?????? |
5
|
{?, ...} |
A?????? |
6
|
{?, ...} |
A?????? |
7
|
{?, ...} |
A?????? |
8
|
{?, ...} |
A?????? |
9
|
{?, ...} |
A?????? |
10
|
{?, ...} |
A?????? |
11
|
{?, ...} |
A?????? |
12
|
{?, ...} |
A?????? |
13
|
{?, ...} |
A?????? |
14
|
{?, ...} |
A?????? |
15
|
{?, ...} |
A?????? |
16
|
{?, ...} |
A?????? |
17
|
{?, ...} |
A?????? |
18
|
{?, ...} |
A?????? |
19
|
{?, ...} |
A?????? |
20
|
{?, ...} |
A?????? |
21
|
{?, ...} |
A?????? |
22
|
{?, ...} |
A?????? |
23
|
{?, ...} |
A?????? |
24
|
{?, ...} |
A?????? |
25
|
{?, ...} |
A?????? |
26
|
{?, ...} |
A?????? |
27
|
{?, ...} |
A?????? |
28
|
{?, ...} |
A?????? |
29
|
{?, ...} |
A?????? |
30
|
{?, ...} |
A?????? |