正整数A×颠倒数A=正整数B×颠倒数B

王守恩

正整数A×颠倒数A=正整数B×颠倒数B。

譬如：144×441=252×252。

譬如：168×861=294×492。

我们不妨约定A是最小的那个。

您也来一个？谢谢！

northwolves

1. f[n_]:=(m=IntegerReverse@n;If[m>=n,{m*n,n,m},{}]);
   
2. s=Select[Table[f[n],{n,10000}],#!={}&];
   
3. t=Select[Gather[s,First@#1==First@#2&],Length@#>1&]// Grid

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{63504,144,441}        {63504,252,252}
{101556,156,651}        {101556,273,372}
{144648,168,861}        {144648,294,492}
{185472,276,672}        {185472,384,483}
{5166504,1224,4221}        {5166504,2142,2412}
{7812756,1236,6321}        {7812756,2163,3612}
{10509408,1248,8421}        {10509408,2184,4812}
{8262306,1326,6231}        {8262306,2613,3162}
{12494209,1339,9331}        {12494209,3193,3913}
{5955264,1344,4431}        {5955264,2352,2532}
{8856036,1356,6531}        {8856036,2373,3732}
{11807208,1368,8631}        {11807208,2394,4932}
{11768148,1428,8241}        {11768148,2814,4182}
{9523696,1456,6541}        {9523696,2743,3472}
{6794424,1464,4641}        {6794424,2562,2652}
{9949716,1476,6741}        {9949716,2583,3852}
{13564768,1568,8651}        {13564768,2954,4592}
{7683984,1584,4851}        {7683984,2772,2772}
{10865686,1586,6851}        {10865686,2873,3782}
{11093796,1596,6951}        {11093796,2793,3972}
{14201712,2316,6132}        {14201712,3504,4053}
{19164096,2328,8232}        {19164096,4074,4704}
{15089472,2346,6432}        {15089472,3264,4623}
{22818208,2369,9632}        {22818208,3296,6923}
{15449112,2436,6342}        {15449112,3624,4263}
{20666016,2448,8442}        {20666016,4284,4824}
{16746912,2556,6552}        {16746912,3744,4473}
{22218336,2568,8652}        {22218336,4494,4944}
{17393152,2576,6752}        {17393152,3584,4853}
{24706318,2639,9362}        {24706318,3926,6293}
{18095112,2676,6762}        {18095112,3864,4683}
{19493712,2796,6972}        {19493712,3984,4893}
{29973924,3468,8643}        {29973924,4386,6834}
{29116584,3528,8253}        {29116584,4716,6174}
{30873024,3648,8463}        {30873024,4836,6384}
{32679864,3768,8673}        {32679864,4956,6594}
{41478016,4439,9344}        {41478016,6176,6716}
{45121216,4669,9664}        {45121216,6496,6946}

northwolves

改成这样就可以了：

1. Union@@@Select[GatherBy[Select[Table[m=IntegerReverse@n;{If[m>=n,m*n,0],n},{n,100000}],#[[1]]>0&],First],Length@#>1&]

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王守恩

northwolves 发表于 2025-2-22 19:32
改成这样就可以了：

{144, 252, 63504}, {156, 273, 101556}, {168, 294, 144648}, {276, 384, 185472}, {1224, 2142, 5166504}, {1236, 2163, 7812756}, {1248, 2184, 10509408}, {1326, 2613, 8262306}, {1344, 2352, 5955264}, {1356, 2373, 8856036},
{1368, 2394, 11807208}, {1428, 2814, 11768148}, {1456, 2743, 9523696}, {1464, 2562, 6794424}, {1476, 2583, 9949716}, {1568, 2954, 13564768}, {1584, 2772, 7683984}, {1586, 2873, 10865686}, {1596, 2793, 11093796}

第3个数是可以去掉的。第2个数是与第1个数有联系, 光凭第1个数怎么把第2个数拉出来？光凭第1个数能把第2个数拉出来吗？再想想。

第2个数也去掉——144, 156, 168, 276, 1224, 1236, 1248, 1326, 1339, 1344, 1356, 1368, 1428, 1456, 1464, 1476, 1568, 1584, 1586, 1596, 2316, 2346, 2369, 2436, 2556, 2576——OEIS没有这串数。挺诱人的！

northwolves

王守恩 发表于 2025-2-23 09:13
{144, 252, 63504}, {156, 273, 101556}, {168, 294, 144648}, {276, 384, 185472}, {1224, 2142, 5166504 ...

1. s = Union @@@
   
2.   Select[GatherBy[
   
3.     Select[Table[
   
4.       m = IntegerReverse@n; {If[m >= n, m*n, 0], n}, {n,
   
5.        100000}], #[[1]] > 0 &], First], Length@# > 1 &]; t =
   
6. s[[All, 1]]

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northwolves

光凭第1个数怎么把第2个数拉出来
-----------------------------------------
筛选即可

1. f[n_] := (m = n*IntegerReverse@n;
   
2.   Select[Range[n + 1, Sqrt[m]], #*IntegerReverse@# == m &]);
   
3. a = {144, 156, 168, 276, 1224, 1236, 1248, 1326, 1339, 1344};
   
4. f[#] & /@ a // Flatten

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{252, 273, 294, 384, 2142, 2163, 2184, 2613, 3193, 2352}

王守恩

northwolves 发表于 2025-2-23 09:54
光凭第1个数怎么把第2个数拉出来
-----------------------------------------
筛选即可

正整数A×颠倒数A=正整数B×颠倒数B。

譬如：144×441=252×252。

譬如：168×861=294×492。

我们不妨约定A是最小的那个。

这串数就是A——144, 156, 168, 276, 1224, 1236, 1248, 1326, 1339, 1344, 1356, 1368, 1428, 1456, 1464, 1476, 1568, 1584, 1586, 1596, 2316, 2346, 2369, 2436, 2556, 2576,

凭这个A——可以有颠倒数A——可以有A×颠倒数A的积——可以有B×颠倒数B的积——可以有B。 光凭A怎么把B拉出来？光凭A能把B拉出来吗？我心里没底。

凭A能把B拉出来。这串数就完整了。题目简单, 思路清晰。可以去OEIS。OEIS没有这串数。挺诱人的！