倒过来写的平方数

王守恩 · 发表于 2025-2-11 09:15:09

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有这么一串数——{1, 4, 9, 18, 46, 52, 61, 63, 94, 121, 144, 148, 163, 169, 423, 441, 484, 487, 522, 526, 652, 675, 676, 691, 925, 927, 961, 982, ......}

规律是这样: 把平方数倒过来写；然后按从小到大重新排列。

问2个问题。

(1), 第2025个数是几？

(2), 5202是第几个数？

王守恩 · 发表于 2025-2-12 12:57:36

a(1)=1=1,
a(2)=2=4,
a(3)=3=9,
a(4)=9=18,
a(5)=8=46,
a(6)=5=52,
a(7)=4=61,
a(8)=6=63,
a(9)=7=94,
a(10)=11=121,
a(11)=21=144,
a(12)=29=148,
a(13)=19=163,
a(14)=31=169,
a(15)=18=423,
a(16)=12=441,
a(17)=22=484,
a(18)=28=487,
a(19)=15=522,
a(20)=25=526,
a(21)=16=652,
a(22)=24=675,
a(23)=26=676,
a(24)=14=691,
a(25)=23=925,
a(26)=27=927,
a(27)=13=961,
a(28)=17=982,

nyy · 发表于 2025-2-12 16:59:42

一万，一千万，这些数都是

王守恩 · 发表于 2025-2-12 18:23:01

这样的1位数有3个——1, 4, 9,

这样的2位数有6个——18, 46, 52, 61, 63, 94,

这样的3位数有19个——121, 144, 148, 163, 169, 423, 441, 484, 487, 522, 526, 652, 675, 676, 691, 925, 927, 961, 982,
这样的4位数有62个——1042, 1062, 1089, 1251, 1273, 1297, 1405, 1426, 1656, 1674, 1828, 1843, 1861, 4032, 4069, 4072, 4201, 4264, 4276, 4441, 4477, 4483, 4633, 4648, 4671, 4806, 4815, 5202, 5203, 5209, 5221,
5224, 5227, 5265, 6112, 6129, 6192, 6313, 6388, 6391, 6507, 6511, 6534, 6745, 6775, 6904, 6921, 6937, 9022, 9049, 9082, 9235, 9295, 9423, 9468, 9481, 9631, 9657, 9693, 9801, 9844, 9886,

这样的5位数有195个——10026, 10036, 10201, 10222, 10404, 10498, 10609, 10693, 10822, 12175, 12186, 12321, 12391, 12538, 12544, 12753, 12769, 12952, 14023, 14292, 14425, 14437, 14641, 14661,

这样的6位数有615个——

得到这样一串数——3, 6, 19, 62, 195, 615, 1946, 6154, 19460, 61540, 194605, 615395, 1946050, 6153950, 19460499, 61539501, 194604990, 615395010, 1946049894, 6153950106, 19460498941, 61539501059, 194604989415, 615395010585,

这通项公式可不好找了！

northwolves · 发表于 2025-2-12 22:31:57

1 3
2 6
3 19
4 62
5 195
6 615
7 1946
8 6154
9 19460
10 61540
11 194605
12 615395
13 1946050
14 6153950
15 19460499
16 61539501
17 194604990
18 615395010
19 1946049894
20 6153950106
21 19460498941
22 61539501059
23 194604989415
24 615395010585
25 1946049894152
26 6153950105848
27 19460498941515
28 61539501058485
29 194604989415154
30 615395010584846
31 1946049894151542
32 6153950105848458
33 19460498941515414
34 61539501058484586
35 194604989415154140
36 615395010584845860
37 1946049894151541398
38 6153950105848458602
39 19460498941515413988
40 61539501058484586012
41 194604989415154139880
42 615395010584845860120
43 1946049894151541398799
44 6153950105848458601201
45 19460498941515413987990
46 61539501058484586012010
47 194604989415154139879901
48 615395010584845860120099
49 1946049894151541398799004
50 6153950105848458601200996
51 19460498941515413987990042
52 61539501058484586012009958
53 194604989415154139879900419
54 615395010584845860120099581
55 1946049894151541398799004190
56 6153950105848458601200995810
57 19460498941515413987990041900
58 61539501058484586012009958100
59 194604989415154139879900418999
60 615395010584845860120099581001
61 1946049894151541398799004189989
62 6153950105848458601200995810011
63 19460498941515413987990041899895
64 61539501058484586012009958100105
65 194604989415154139879900418998944
66 615395010584845860120099581001056
67 1946049894151541398799004189989447
68 6153950105848458601200995810010553
69 19460498941515413987990041899894467
70 61539501058484586012009958100105533
71 194604989415154139879900418998944668
72 615395010584845860120099581001055332
73 1946049894151541398799004189989446680
74 6153950105848458601200995810010553320
75 19460498941515413987990041899894466804
76 61539501058484586012009958100105533196
77 194604989415154139879900418998944668034
78 615395010584845860120099581001055331966
79 1946049894151541398799004189989446680348
80 6153950105848458601200995810010553319652
81 19460498941515413987990041899894466803476
82 61539501058484586012009958100105533196524
83 194604989415154139879900418998944668034760
84 615395010584845860120099581001055331965240
85 1946049894151541398799004189989446680347600
86 6153950105848458601200995810010553319652400
87 19460498941515413987990041899894466803475996
88 61539501058484586012009958100105533196524004
89 194604989415154139879900418998944668034759962
90 615395010584845860120099581001055331965240038
91 1946049894151541398799004189989446680347599626
92 6153950105848458601200995810010553319652400374
93 19460498941515413987990041899894466803475996254
94 61539501058484586012009958100105533196524003746
95 194604989415154139879900418998944668034759962539
96 615395010584845860120099581001055331965240037461
97 1946049894151541398799004189989446680347599625393
98 6153950105848458601200995810010553319652400374607
99 19460498941515413987990041899894466803475996253927
100 61539501058484586012009958100105533196524003746073

northwolves · 发表于 2025-2-12 22:47:47

a[n_]:=Sum[(1+4*Floor[k/4]-2Floor[k/2])*Ceiling@Sqrt[10^(n+1-k)],{k,6}];Table[{n,a[n]},{n,100}]//TableForm

复制代码

a[n_]:=Sum[Ceiling@Sqrt[10^(n-k)]*{1,-1,-1,1,1,-1}[[k+1]],{k,0,5}];Table[{n,a[n]},{n,100}]//TableForm

复制代码

northwolves · 发表于 2025-2-12 22:52:16

$a_n=\sum _{k=1}^6 (1+4\lfloor \frac{k}{4}\rfloor -2\lfloor \frac{k}{2}\rfloor)\ceil \sqrt{10^{n+1-k}}$

王守恩 · 发表于 2025-2-13 07:29:40

northwolves 发表于 2025-2-12 22:31
1 3
2 6
3 19

这可是一串OEIS没有的数——虽然有缺陷。

{6, 19, 62, 195, 615, 1946, 6154, 19460, 61540, 194605, 615395, 1946050, 6153950, 19460499, 61539501, 194604990, 615395010, 1946049894, 6153950106, 19460498941, 61539501059, 194604989415, 615395010585, 1946049894152}

Table[Floor[10^(k/2)] - Floor[10^((k - 1)/2)] - Floor[10^((k - 2)/2)] + Floor[10^((k - 3)/2)], {k, 2, 25}]

复制代码

王守恩 · 发表于 2025-2-13 07:31:51

总算引出来了！谢谢！！

A074896——是有这串数——{1, 4, 9, 18, 46, 52, 61, 63, 94, 121, 144, 148, 163, 169, 423, 441, 484, 487, 522, 526, 652, 675, 676, 691, 925, 927, 961, 982, ......}

A074896——可惜没有通项公式——或是“可以复制粘贴的代码”没有。也就是说这2个问题解决不了！

(1), 第2025个数是几？

(2), 5202是第几个数？

northwolves · 发表于 2025-2-13 08:09:25

(2), 5202是第几个数？
----------------------------
这个很简单啊，1-45逆转，去掉10的倍数，第41个

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[提问] 倒过来写的平方数

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