三圆相切的解法难题

iseemu2009

三圆相切的解法难题

王守恩

三个圆的半径都是整数时,三个半径之和——{9, 18, 27, 36, 45, 49, 54, 63, 72, 81, 98, 147, 169, 196, 245, 294, 338, 343, 392, 441, 507, 676, 845, 882, 961,
1014, 1183, 1323, 1352, 1521, 1764, 1849, 1922, 2205, 2646, 2883, 3087, 3249, 3528, 3698, 3844, 3969, 4805, 5329, 5547, 5766, 6498,

iseemu2009

王守恩 发表于 2025-3-8 10:21
题目=这串数？——{9, 18, 27, 36, 45, 49, 54, 63, 72, 81, 98, 147, 169, 196, 245, 294, 338, 343, 392, ...

题目就是任意指定一个正整数 n ，满足 0≤ r1≤r2≤n，然后根据图中几何关系求出 r3，要求三个圆的半径都是整数时，求三个半径之和是多少。

iseemu2009

r3的半径

northwolves

$\frac{1}{\sqrt{r_3}}=\frac{1}{\sqrt{r_1}}+\frac{1}{\sqrt{r_2}$

iseemu2009

配对数

iseemu2009

1. Clear["`*"]
   
2. n = 5;
   
3. a = Flatten[Table[{i, j}, {j, n}, {i, j}], 1];
   
4. b = Table[
   
5.    IntegerQ[(a[[i, 1]]  a[[i, 2]])/(
   
6.     a[[i, 1]] + a[[i, 2]] + 2 Sqrt[a[[i, 1]] a[[i, 2]]])], {i,
   
7.     Length[a]}];
   
8. c = Count[b, True];
   
9. d = Position[b, True];
   
10. e = a[[Flatten[d]]];
    
11. f = Table[(a[[i, 1]]  a[[i, 2]])/(
    
12.    a[[i, 1]] + a[[i, 2]] + 2 Sqrt[a[[i, 1]] a[[i, 2]]]), {i, d}];
    
13. g = Total[Flatten[e, 3]] + Total[Flatten[f, 3]]

复制代码

iseemu2009

上面是我编的mathematica程序，n 的值可以任意变化。可以验证 S(5)=9， S(10)=27， S(20)=135，S(100)=3072，S(1234)=475425。

northwolves

s(1234)=475425

northwolves

1. f[n_]:=(s={};For[r3=1,r3<=n/4,r3++,For[r2=4*r3,r2<=n,r2++,r1=r2 r3/(r2+r3-2Sqrt[r2 r3]);If [IntegerQ[r1]&& r1<=r2,AppendTo[s,{r3,r1,r2}]]]];{Total@Flatten@s,Length@s,s});f@100
   
2. f@1234

复制代码

{3072,27,{{1,4,4},{2,8,8},{3,12,12},{4,16,16},{4,9,36},{5,20,20},{6,24,24},{7,28,28},{8,32,32},{8,18,72},{9,36,36},{10,40,40},{11,44,44},{12,48,48},{13,52,52},{14,56,56},{15,60,60},{16,64,64},{17,68,68},{18,72,72},{19,76,76},{20,80,80},{21,84,84},{22,88,88},{23,92,92},{24,96,96},{25,100,100}}}

{475425,361,{{1,4,4},{2,8,8},{3,12,12},{4,16,16},{4,9,36},{5,20,20},{6,24,24},{7,28,28},{8,32,32},{8,18,72},{9,36,36},{9,16,144},{10,40,40},{11,44,44},{12,48,48},{12,27,108},{13,52,52},{14,56,56},{15,60,60},{16,64,64},{16,36,144},{16,25,400},{17,68,68},{18,72,72},{18,32,288},{19,76,76},{20,80,80},{20,45,180},{21,84,84},{22,88,88},{23,92,92},{24,96,96},{24,54,216},{25,100,100},{25,36,900},{26,104,104},{27,108,108},{27,48,432},{28,112,112},{28,63,252},{29,116,116},{30,120,120},{31,124,124},{32,128,128},{32,72,288},{32,50,800},{33,132,132},{34,136,136},{35,140,140},{36,144,144},{36,100,225},{36,81,324},{36,64,576},{37,148,148},{38,152,152},{39,156,156},{40,160,160},{40,90,360},{41,164,164},{42,168,168},{43,172,172},{44,176,176},{44,99,396},{45,180,180},{45,80,720},{46,184,184},{47,188,188},{48,192,192},{48,108,432},{48,75,1200},{49,196,196},{50,200,200},{51,204,204},{52,208,208},{52,117,468},{53,212,212},{54,216,216},{54,96,864},{55,220,220},{56,224,224},{56,126,504},{57,228,228},{58,232,232},{59,236,236},{60,240,240},{60,135,540},{61,244,244},{62,248,248},{63,252,252},{63,112,1008},{64,256,256},{64,144,576},{65,260,260},{66,264,264},{67,268,268},{68,272,272},{68,153,612},{69,276,276},{70,280,280},{71,284,284},{72,288,288},{72,200,450},{72,162,648},{72,128,1152},{73,292,292},{74,296,296},{75,300,300},{76,304,304},{76,171,684},{77,308,308},{78,312,312},{79,316,316},{80,320,320},{80,180,720},{81,324,324},{82,328,328},{83,332,332},{84,336,336},{84,189,756},{85,340,340},{86,344,344},{87,348,348},{88,352,352},{88,198,792},{89,356,356},{90,360,360},{91,364,364},{92,368,368},{92,207,828},{93,372,372},{94,376,376},{95,380,380},{96,384,384},{96,216,864},{97,388,388},{98,392,392},{99,396,396},{100,400,400},{100,225,900},{100,196,1225},{101,404,404},{102,408,408},{103,412,412},{104,416,416},{104,234,936},{105,420,420},{106,424,424},{107,428,428},{108,432,432},{108,300,675},{108,243,972},{109,436,436},{110,440,440},{111,444,444},{112,448,448},{112,252,1008},{113,452,452},{114,456,456},{115,460,460},{116,464,464},{116,261,1044},{117,468,468},{118,472,472},{119,476,476},{120,480,480},{120,270,1080},{121,484,484},{122,488,488},{123,492,492},{124,496,496},{124,279,1116},{125,500,500},{126,504,504},{127,508,508},{128,512,512},{128,288,1152},{129,516,516},{130,520,520},{131,524,524},{132,528,528},{132,297,1188},{133,532,532},{134,536,536},{135,540,540},{136,544,544},{136,306,1224},{137,548,548},{138,552,552},{139,556,556},{140,560,560},{141,564,564},{142,568,568},{143,572,572},{144,576,576},{144,441,784},{144,400,900},{145,580,580},{146,584,584},{147,588,588},{148,592,592},{149,596,596},{150,600,600},{151,604,604},{152,608,608},{153,612,612},{154,616,616},{155,620,620},{156,624,624},{157,628,628},{158,632,632},{159,636,636},{160,640,640},{161,644,644},{162,648,648},{163,652,652},{164,656,656},{165,660,660},{166,664,664},{167,668,668},{168,672,672},{169,676,676},{170,680,680},{171,684,684},{172,688,688},{173,692,692},{174,696,696},{175,700,700},{176,704,704},{177,708,708},{178,712,712},{179,716,716},{180,720,720},{180,500,1125},{181,724,724},{182,728,728},{183,732,732},{184,736,736},{185,740,740},{186,744,744},{187,748,748},{188,752,752},{189,756,756},{190,760,760},{191,764,764},{192,768,768},{193,772,772},{194,776,776},{195,780,780},{196,784,784},{197,788,788},{198,792,792},{199,796,796},{200,800,800},{201,804,804},{202,808,808},{203,812,812},{204,816,816},{205,820,820},{206,824,824},{207,828,828},{208,832,832},{209,836,836},{210,840,840},{211,844,844},{212,848,848},{213,852,852},{214,856,856},{215,860,860},{216,864,864},{217,868,868},{218,872,872},{219,876,876},{220,880,880},{221,884,884},{222,888,888},{223,892,892},{224,896,896},{225,900,900},{226,904,904},{227,908,908},{228,912,912},{229,916,916},{230,920,920},{231,924,924},{232,928,928},{233,932,932},{234,936,936},{235,940,940},{236,944,944},{237,948,948},{238,952,952},{239,956,956},{240,960,960},{241,964,964},{242,968,968},{243,972,972},{244,976,976},{245,980,980},{246,984,984},{247,988,988},{248,992,992},{249,996,996},{250,1000,1000},{251,1004,1004},{252,1008,1008},{253,1012,1012},{254,1016,1016},{255,1020,1020},{256,1024,1024},{257,1028,1028},{258,1032,1032},{259,1036,1036},{260,1040,1040},{261,1044,1044},{262,1048,1048},{263,1052,1052},{264,1056,1056},{265,1060,1060},{266,1064,1064},{267,1068,1068},{268,1072,1072},{269,1076,1076},{270,1080,1080},{271,1084,1084},{272,1088,1088},{273,1092,1092},{274,1096,1096},{275,1100,1100},{276,1104,1104},{277,1108,1108},{278,1112,1112},{279,1116,1116},{280,1120,1120},{281,1124,1124},{282,1128,1128},{283,1132,1132},{284,1136,1136},{285,1140,1140},{286,1144,1144},{287,1148,1148},{288,1152,1152},{289,1156,1156},{290,1160,1160},{291,1164,1164},{292,1168,1168},{293,1172,1172},{294,1176,1176},{295,1180,1180},{296,1184,1184},{297,1188,1188},{298,1192,1192},{299,1196,1196},{300,1200,1200},{301,1204,1204},{302,1208,1208},{303,1212,1212},{304,1216,1216},{305,1220,1220},{306,1224,1224},{307,1228,1228},{308,1232,1232}}}