有这样一串数(OEIS找不到) - 第4页 - 刨根究底 - 数学研发论坛

northwolves 发表于 2024-5-12 23:18:09

N=201499416970855328645120000={1,2,3,4,5,6,21,25,26,27,28,30,32,33,34,38,42,44,46,58} {7,8,9,10,11,12,13,14,15,16,17,18,19,20,22,23,24,29,35,36}

northwolves 发表于 2024-5-12 23:28:08

N=2174970848542766432256000000={1,2,3,4,5,6,21,25,26,27,28,30,32,33,34,38,40,42,46,55,58} {7,8,9,10,11,12,13,14,15,16,17,18,19,20,22,23,24,29,35,36,50}

王守恩 发表于 2024-5-13 07:49:41

N=02, 02*03=06,
N=03, N(02)*(06.0)(3*2)=36,
N=04, N(03)*(06.7)(5*4/3)=240,
N=05, N(04)*(10.5)(7*3/2)=2520,
N=06, N(05)*(12.0)(4*3)=30240,
N=07, N(06)*(14.7)(11*4/3)=443520,
N=08, N(07)*(15.0)(5*3)=6652800,
N=09, N(08)*(20.0)(5*4)=133056000,
N=10, N(09)*(15.6)(13*3*2/5)=2075673600,
N=11, N(10)*(28.0)(7*4)=58118860800,
N=12, N(11)*(21.9)(17*9/7)=1270312243200,
N=13, N(12)*(23.3)(7*5*2/3)=29640619008000,
N=14, N(13)*(28.5)(19*3/2)=844757641728000,
N=15, N(14)*(30.0)(5*3*2)=25342729251840000,
N=16, N(15)*(32.0)(32)=810967336058880000,
N=17, N(16)*(34.5)(23*3/2)=27978373094031360000,
N=18, N(17)*(38.5)(11*7/2)=1077167364120207360000,
N=19, N(18)*(40.0)(5*8)=43086694564808294400000,
N=20, N(19)*(34.8)(29*3*2/5)=1499416970855328645120000,
N=21, N(20)*(50.0)(25*2)=74970848542766432256000000,

northwolves 发表于 2024-5-13 09:28:58

n=22, 2788915565790911279923200000= {1, 2, 3, 4, 5, 6, 21, 22, 26, 27, 28, 30, 34, 35, 36, 38, 40, 44, 46, 58, 62} {7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 23, 24, 25, 29, 31, 32, 33, 63}

mathe 发表于 2024-5-13 09:35:27

Result for A354457 & C++ Code

N=2
6
{ 2 3 }{ 1 6 }
N=3
36
{ 2 3 6 }{ 1 4 9 }
N=4
240
{ 2 4 5 6 }{ 1 3 8 10 }
N=5
2520
{ 3 4 5 6 7 }{ 1 2 9 10 14 }
N=6
30240
{ 2 5 6 7 8 9 }{ 1 3 4 12 14 15 }
N=7
443520
{ 3 4 5 6 7 8 22 }{ 1 2 9 10 11 14 16 }
N=8
6652800
{ 4 5 6 7 8 9 10 11 }{ 1 2 3 12 14 15 20 22 }
N=9
133056000
{ 2 5 7 8 9 10 11 12 20 }{ 1 3 4 6 14 15 16 22 25 }
N=10
2075673600
{ 2 6 7 8 9 10 11 12 13 20 }{ 1 3 4 5 14 15 16 18 22 26 }
N=11
58118860800
{ 2 7 8 9 10 11 12 13 14 15 16 }{ 1 3 4 5 6 20 21 22 24 26 28 }
N=12
1270312243200
{ 1 8 9 10 11 12 13 14 15 16 17 18 }{ 2 3 4 5 6 7 20 22 24 26 27 34 }
N=13
29640619008000
{ 3 7 8 9 10 11 12 13 14 15 16 17 20 }{ 1 2 4 5 6 18 21 22 24 25 26 28 34 }
N=14
844757641728000
{ 2 7 9 10 11 12 13 14 15 16 17 18 19 20 }{ 1 3 4 5 6 8 21 22 25 26 27 28 34 38 }
N=15
25342729251840000
{ 4 5 9 10 11 12 13 14 15 16 17 18 19 20 21 }{ 1 2 3 6 7 8 22 24 25 26 27 30 34 35 38 }
N=16
810967336058880000
{ 5 6 8 10 11 12 13 14 15 16 17 18 19 20 21 24 }{ 1 2 3 4 7 9 22 25 26 27 28 30 32 34 38 40 }
N=17
27978373094031360000
{ 4 5 9 10 12 13 14 15 16 17 18 19 20 21 22 23 24 }{ 1 2 3 6 7 8 11 25 26 27 30 32 34 35 36 38 46 }
N=18
1077167364120207360000
{ 1 9 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 28 }{ 2 3 4 5 6 7 8 10 26 27 30 33 34 35 38 42 44 46 }
N=19
43086694564808294400000
{ 4 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 28 }{ 1 2 3 5 6 7 8 26 27 30 32 33 34 35 38 42 44 46 50 }
N=20
1499416970855328645120000
{ 1 8 11 12 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 }{ 2 3 4 5 6 7 9 10 13 30 32 33 34 35 36 38 42 44 46 58 }
N=21
74970848542766432256000000
{ 1 5 11 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 32 }{ 2 3 4 6 7 8 9 10 12 13 33 34 35 38 40 42 44 45 46 50 58 }
N=22
2788915565790911279923200000
{ 1 2 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 }{ 3 4 5 6 7 8 9 10 11 12 13 34 35 36 38 40 42 44 45 46 58 62 }
N=23
140345428472058761183232000000
{ 2 10 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 32 33 }{ 1 3 4 5 6 7 8 9 11 34 35 36 38 39 40 42 44 45 46 48 50 52 58 }
N=24
6090991595687350235352268800000
{ 4 7 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 }{ 1 2 3 5 6 8 9 10 11 34 35 36 38 39 40 42 44 45 46 48 49 52 58 62 }
N=25
321952412914902798154334208000000
{ 2 8 12 13 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 35 37 }{ 1 3 4 5 6 7 9 10 11 14 34 36 38 39 40 42 44 45 46 48 50 52 58 62 74 }
N=26
18029335123234556696642715648000000
{ 1 4 13 14 16 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 40 }{ 2 3 5 6 7 8 9 10 11 12 15 17 38 39 42 44 45 46 48 49 50 52 58 62 64 74 }
N=27
1103395309541954869834534197657600000
{ 2 8 12 14 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 39 }{ 1 3 4 5 6 7 9 10 11 13 15 38 40 42 44 45 46 48 49 51 52 54 58 62 64 68 74 }
N=28
56549009614025187079019877629952000000
{ 1 2 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 39 40 41 42 }{ 3 4 5 6 7 8 9 10 11 12 13 14 15 38 44 45 46 48 49 50 51 52 54 58 62 68 74 82 }
N=29
3641204521488451070453962852270080000000
{ 1 4 15 16 17 18 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 42 44 }{ 2 3 5 6 7 8 9 10 11 12 13 14 19 45 46 48 49 50 51 52 54 55 58 60 62 64 66 68 74 }
N=30
204255022725858975729419797999386624000000
{ 1 6 14 16 17 18 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 }{ 2 3 4 5 7 8 9 10 11 12 13 15 19 44 45 46 48 49 50 51 52 54 56 58 62 63 68 74 82 86 }
N=31
12540308372006225486643448063218155520000000
{ 1 8 14 16 17 18 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 44 45 }{ 2 3 4 5 6 7 9 10 11 12 13 15 19 46 48 49 50 51 52 54 55 56 58 60 62 63 64 66 68 74 82 }
N=32
808849889994401543888502400077571031040000000
{ 1 4 14 17 18 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 48 }{ 2 3 5 6 7 8 9 10 11 12 13 15 16 19 46 49 50 51 52 54 55 56 58 60 62 63 66 68 72 74 82 86 }
N=33
50687926439649163417012817071527784611840000000
{ 1 6 14 16 17 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 47 48 }{ 2 3 4 5 7 8 9 10 11 12 13 15 18 19 46 49 50 51 52 54 55 56 58 60 62 63 64 66 68 74 82 86 94 }

Number of solutions

northwolves 发表于 2024-5-13 09:37:43

只是参考mathe的构造法半手工半编程得到的数字，不知如何验证是否最小。

好容易觉得是最小的，拆解移动几次，就会发现一个更小点的。没有最小，只有更小。

mathe 发表于 2024-5-13 10:09:52

还有A354697基本上可以用类似方案解决
C++代码
思路很简单，就是从小到大枚举所有2N个整数的乘积，为了枚举2N个整数的集合，对于任意一个候选集合$\{a_1,a_2,...,a_{2N}\}$,
我们寻找最大的j,使得$a_{j+1}-a_j \ge 2, a_{2N}-a_{2N-1}=a_{2N-1}-a_{2N-2}=\cdots=a_{j+2}-a_{j+1}=1$
这时如果我们将$a_{j+1}$修改为$a_{j+1}-1$,那么可以得到一个乘积更小的2N元集合，看成上一个2N元集合父节点。
通过这种修改关系，我们可以将所有的2N元集合构成一棵树，其中对于A354457，根节点就是集合{1,2,3,...,2N}; 对于A354697，根节点为集合{2,3,...,2N,2N+1},其它没有任何区别。
而这种方案每个节点最多两个孩子节点。我们只要将所有节点根据乘积从小到大排列，然后每次将最小节点产生的两个孩子节点插入这个有序序列即可，直到当前最小节点的乘积是一个完全平方数。

mathe 发表于 2024-5-13 12:05:40

A354697:
2 12
3 120
4 720
5 10080
6 110880
7 1814400
8 26611200
9 518918400
10 10378368000
11 261534873600
12 5928123801600
13 168951528345600
14 4505374089216000
15 152056375511040000
16 4663062182338560000
17 167870238564188160000
18 6463004184721244160000
19 249902828475888107520000
20 10495918795987300515840000
21 449825091256598593536000000
22 19522408960536378959462400000
23 870141656526764319336038400000
24 45682436967655126765142016000000
25 2253666890404319587080339456000000
26 137924413692744358729316774707200000
27 7761628770552476657904689086464000000
28 452392076912201496632159021039616000000

王守恩 发表于 2024-5-13 12:55:35

mathe 发表于 2024-5-13 12:05
A354697:
2 12
3 120

再来一串？
N=2,
15,
{3, 5}{1, 15}
N=3,
405,
{3, 5, 27}{1, 9, 45}
N=4,
2835,
{3, 5, 7, 27}{1, 9, 15, 21}
N=5,
93555,
{3, 5, 9, 21, 33}{1, 7, 11, 15, 81}
N=6,

northwolves 发表于 2024-5-13 13:18:24

本帖最后由 northwolves 于 2024-5-13 13:47 编辑

n=3,

225 {3, 5, 15}{1, 9, 25}

n=4

2835 {1,5,21,27} {3,7,9,15}
2835 {1,7,15,27} {3,5,9,21}
2835 {1,9,15,21} {3,5,7,27}

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