A^2有4432个结果，以最高两位98为例：
Select,{8}],Times@@Tally][]==2^10&]
{9801753624,9801763524,9804317625,9804631275,9804652317,9804731652,9806473512,9806732514,9807213465,9807653421,9814562370,9814753620,9815763402,9816254073,9816452073,9817542063,9817603542,9820156743,9820536417,9820547631,9821365740,9821403567,9823407516,9824037165,9824513670,9824705613,9824710356,9825413670,9825640137,9827041635,9827456013,9827650341,9827650431,9830127645,9830167425,9832140567,9834652071,9835016427,9835724160,9837621045,9840312576,9840567213,9842573106,9843061752,9845371062,9846213057,9847365012,9847561032,9852763014,9853024671,9853210647,9853267140,9853426017,9856073124,9857064132,9857162430,9857246103,9857324601,9857360421,9861034572,9863245017,9863415207,9864527013,9865017243,9865403721,9867521340,9871054236,9871065423,9874123560,9875146302,9876104253,9876124053,9876153024}
本帖最后由 王守恩 于 2024-2-25 18:12 编辑
数字串T(n): x*x各个数位上的数码和=9*T。求最小的T(n)。
数字串A(n): x*x各个数位上的数码和=9(x+A)。求最小的A(n)。
z=x*x的位数，T<45,z=正确值，T>44,z=估算值。
估算值=Round[(n*36)/33]
T = x + A,2*x - y=z
01=01+00, 2*01-1=01,
02=02+00, 2*02-1=03,
03=02+01, 2*02-0=04,
04=03+01, 2*03-1=05,
05=04+01, 2*04-1=07,
06=04+02, 2*04-1=07,
07=04+03, 2*04-0=08,
08=05+03, 2*05-0=10,
09=05+04, 2*05-0=10,
10=06+04, 2*06-0=12,
11=07+04, 2*07-1=13,
12=07+05, 2*07-0=14,
13=08+05, 2*08-1=15,
14=08+06, 2*08-0=16,
15=09+06, 2*09-1=17,
16=09+07, 2*09-0=18,
17=10+07, 2*10-1=19,
18=10+08, 2*10-0=20,
19=11+08, 2*11-1=21,
20=11+09, 2*11-0=22,
21=12+09, 2*12-0=24,
22=13+09, 2*13-1=25,
23=13+10, 2*13-0=26,
24=14+10, 2*14-1=27,
25=14+11, 2*14-0=28,
26=15+11, 2*15-1=29,
27=15+12, 2*15-0=30,
28=16+12, 2*16-1=31,
29=16+13, 2*16-0=32,
30=17+13, 2*17-0=34,
31=18+13, 2*18-1=35,
32=18+14, 2*18-0=36,
33=18+15, 2*18-0=36,
34=19+15, 2*19-0=38,
35=20+15, 2*20-1=39,
36=20+16, 2*20-0=40,
37=21+16, 2*21-1=41,
38=21+17, 2*21-0=42,
39=22+17, 2*22-1=43,
40=22+18, 2*22-0=44,
41=23+18, 2*23-0=46,
42=23+19, 2*23-0=46,
43=24+19, 2*24-1=47,
44=24+20, 2*24-0=48,
45=25+20, 2*25-1=49,
46=25+21, 2*25-0=50,
47=26+21, 2*26-1=51,
48=26+22, 2*26-0=52,
49=27+22, 2*27-1=53,
50=28+22, 2*28-1=55,
51=28+23, 2*28-0=56,
52=29+23, 2*29-1=57,
53=29+24, 2*29-0=58,
54=30+24, 2*30-1=59,
55=30+25, 2*30-0=60,
56=31+25, 2*31-1=61,
57=31+26, 2*31-0=62,
58=32+26, 2*32-1=63,
59=32+27, 2*32-0=64,
60=33+27, 2*33-1=65,
61=34+27, 2*34-1=67,
62=34+28, 2*34-0=68,
63=35+28, 2*35-1=69,
64=35+29, 2*35-0=70,
65=36+29, 2*36-1=71,
66=36+30, 2*36-0=72,
67=37+30, 2*37-1=73,
68=37+31, 2*37-0=74,
69=38+31, 2*38-1=75,
70=38+32, 2*38-0=76,
71=39+32, 2*39-1=77,
72=40+32, 2*40-1=79,
73=40+33, 2*40-0=80,
74=41+33, 2*41-1=81,
75=41+34, 2*41-0=82,
76=42+34, 2*42-1=83,
77=42+35, 2*42-0=84,
78=43+35, 2*43-1=85,
79=43+36, 2*43-0=86,
80=44+36, 2*44-1=87,
81=44+37, 2*44-0=88,
82=45+37, 2*45-1=89,
83=46+37, 2*46-1=91,
84=46+38, 2*46-0=92,
85=47+38, 2*47-1=93,
86=47+39, 2*47-0=94,
87=48+39, 2*48-1=95,
88=48+40, 2*48-0=96,
89=49+40, 2*49-1=97,
90=49+41, 2*49-0=98,
91=50+41, 2*50-1=99,
92=50+42, 2*50-0=00,
93=51+42, 2*51-1=01,
94=52+42, 2*52-1=03,
95=52+43, 2*52-0=04,
96=53+43, 2*53-1=05,
97=53+44, 2*53-0=06,
98=54+44, 2*54-1=07,
99=54+45, 2*54-0=08,
T(n)T=Round[(n*36)/33]
T(n) -，T(n)+由46#数字串(1), 数字串(2)来。
T(n)T=9k+0,T(n) -=9k-1,9k-2,9k-3,9k-4，T(n)+=9k+1,9k+2,9k+3,9k+4,
T(01) -=
T(01)T=01
T(01)+=
T(02) -=
T(02)T=02
T(02)+=
T(03) -=
T(03)T=03
T(03)+=
T(04) -=
T(04)T=04
T(04)+=06
T(05) -=
T(05)T=05
T(05)+=
T(06) -=
T(06)T=07
T(06)+=
T(07) -=
T(07)T=08
T(07)+=
T(08) -=
T(08)T=09
T(08)+=10
T(09) -=
T(09)T=10
T(09)+=
T(10) -=11
T(10)T=11
T(10)+=
T(11) -=
T(11)T=12
T(11)+=
T(12) -=14
T(12)T=13
T(12)+=
T(13) -=
T(13)T=14
T(13)+=15
T(14) -=
T(14)T=15
T(14)+=
T(15) -=
T(15)T=16
T(15)+=18
T(16) -=
T(16)T=17
T(16)+=
T(17) -=
T(17)T=19
T(17)+=19
T(18) -=
T(18)T=20
T(18)+=
T(19) -=
T(19)T=21
T(19)+=22
T(20) -=
T(20)T=22
T(20)+=
T(21) -=23
T(21)T=23
T(21)+=
T(22) -=
T(22)T=24
T(22)+=
T(23) -=26
T(23)T=25
T(23)+=
T(24) -=
T(24)T=26
T(24)+=27
T(25) -=
T(25)T=27
T(25)+=
T(26) -=
T(26)T=28
T(26)+=30
T(27) -=
T(27)T=29
T(27)+=
T(28) -=
T(28)T=31
T(28)+=31
T(29) -=
T(29)T=32
T(29)+=
T(30) -=
T(30)T=33
T(30)+=34
T(31) -=
T(31)T=34
T(31)+=
T(32) -=35
T(32)T=35
T(32)+=
T(33) -=
T(33)T=36
T(33)+=
T(34) -=38
T(34)T=37
T(34)+=
T(35) -=
T(35)T=38
T(35)+=39
T(36) -=
T(36)T=39
T(36)+=
T(37) -=
T(37)T=40
T(37)+=42
T(38) -=
T(38)T=41
T(38)+=
T(39) -=
T(39)T=43
T(39)+=43
T(40) -=
T(40)T=44
T(40)+=
T(41) -=
T(41)T=45
T(41)+=46
T(42) -=
T(42)T=46
T(42)+=
T(43) -=47
T(43)T=47
T(43)+=
T(44) -=
T(44)T=48
T(44)+=
T(45) -=50
T(45)T=49
T(45)+=
T(46) -=
T(46)T=50
T(46)+=51
T(47) -=
T(47)T=51
T(47)+=
T(48) -=
T(48)T=52
T(48)+=54
T(49) -=
T(49)T=53
T(49)+=
T(50) -=
T(50)T=55
T(50)+=55
T(51) -=
T(51)T=56
T(51)+=
T(52) -=
T(52)T=57
T(52)+=58
T(53) -=
T(53)T=58
T(53)+=
T(54) -=59
T(54)T=59
T(54)+=
T(55) -=
T(55)T=60
T(55)+=
T(56) -=62
T(56)T=61
T(56)+=
T(57) -=
T(57)T=62
T(57)+=63
T(58) -=
T(58)T=63
T(58)+=
T(59) -=
T(59)T=64
T(59)+=66
T(60) -=
T(60)T=65
T(60)+=
T(61) -=
T(61)T=67
T(61)+=67
T(62) -=
T(62)T=68
T(62)+=
T(63) -=
T(63)T=69
T(63)+=70
T(64) -=
T(64)T=70
T(64)+=
T(65) -=71
T(65)T=71
T(65)+=
T(66) -=
T(66)T=72
T(66)+=
T(67) -=74
T(67)T=73
T(67)+=
T(68) -=
T(68)T=74
T(68)+=75
T(69) -=
T(69)T=75
T(69)+=
T(70) -=
T(70)T=76
T(70)+=78
T(71) -=
T(71)T=77
T(71)+=
T(72) -=
T(72)T=79
T(72)+=79
T(73) -=
T(73)T=80
T(73)+=
T(74) -=
T(74)T=81
T(74)+=82
T(75) -=
T(75)T=82
T(75)+=
T(76) -=83
T(76)T=83
T(76)+=
T(77) -=
T(77)T=84
T(77)+=
T(78) -=86
T(78)T=85
T(78)+=
T(79) -=
T(79)T=86
T(79)+=87
T(80) -=
T(80)T=87
T(80)+=
T(81) -=
T(81)T=88
T(81)+=90
T(82) -=
T(82)T=89
T(82)+=
T(83) -=
T(83)T=91
T(83)+=91
T(84) -=
T(84)T=92
T(84)+=
T(85) -=
T(85)T=93
T(85)+=94
T(86) -=
T(86)T=94
T(86)+=
T(87) -=95
T(87)T=95
T(87)+=
T(88) -=
T(88)T=96
T(88)+=
T(89) -=98
T(89)T=97
T(89)+=
T(90) -=
T(90)T=98
T(90)+=99
T(91) -=
T(91)T=99
T(91)+=
T(92) -=
T(92)T=00
T(92)+=02
T(93) -=
T(93)T=01
T(93)+=
T(94) -=
T(94)T=03
T(94)+=03
T(95) -=
T(95)T=04
T(95)+=
T(96) -=
T(96)T=05
T(96)+=06
T(97) -=
T(97)T=06
T(97)+=
T(98) -=07
T(98)T=07
T(98)+=
T(99) -=
T(99)T=08
T(99)+=
多说两句？
若T(n)=偶数，则x=T(n)/2, A=n-T(n)/2。
若T(n)=奇数，则x=(T(n)+1)/2, A=n-(T(n)+1)/2。
若T(n) -,T(n)+=偶数，则T(n) -,T(n)+由数字串(1)来。
若T(n) -,T(n)+=奇数，则T(n) -,T(n)+由数字串(2)来。
Python 的整数类型有无限精度，正好用来做这个题。
附件是 Python 代码和计算结果 (n = 1 ~ 2048) :
A 是由 n 个相同数码组成的 n 位数，
A * A 各个数位上的数码和 = 9 * n ，
n = 1 ~ 2024，这样的 A 有多少个？
答案：8991 个。
2024 * 9 = 18216
8991 / 18216 = 49.36%
northwolves 发表于 2024-2-21 15:34
a(162)>123000000000
153 1001283601183
修改前面的代码，并且合并两个代码最优结果，可以得到如下结果(同样两边搜索长度之和选择为41)
100 10954437
101 5477133
102 9272917
103 100427083
104 13415614
105 9893887
106 102419626
107 19672313
108 103860387
109 20736417
110 24060133
111 1001798327
112 104258333
113 29983327
114 1003487917
115 113578114
116 44271886
117 129984576
118 82395387
119 60827617
120 1004489417
121 164277083
122 99477133
123 1034407076
124 197222917
125 1035857133
126 260191833
127 197483417
128 10028903114
129 1140086833
130 434738887
131 10029954614
132 1224291587
133 529027313
134 1260872313
135 706399164
136 100044889864
137 10049328286
138 1664331667
139 994927133
140 10343113114
141 2641873937
142 10237040583
143 2428989417
144 2983284917
145 100344331667
146 10945318633
147 5383305583
148 100497756164
149 13038404764
150 8882454614
151 17204034117
152 9380293167
153 1000494027313
154 103382242126
155 21213179863
156 1001790341833
157 109402911113
158 31144643167
159 104345531667
160 28105157886
161 59999833333
162 1004380206833
163 109452720386
164 69999835714
165 1004484937567
166 134123072917
167 94286790167
168 109544510583
169 99497231067
170 10004818333633
171 1024695027313
172 240827299117
173 10018981984063
174 1090825328813
175 244272388937
176 1174223143167
177 538479339417
178 806207101183
179 10049371516667
180 1256975731667
181 926174913167
182 10094057602664
183 1296109172867
184 10245965044833
185 1974763271886
186 2213594339417
187 100133361063583
188 10435511007133
189 3144835432874
190 100049936529614
191 11175821177837
192 3160522105583
193 11313706234917
194 4472135831667
195 5474468010583
196 100483530987917
197 12193805763583
198 9892825177313
199 100993360672867
200 13740377720386
201 102853244965824
202 14106593458167
203 22108798881667
204 1001997947527837
205 103917751072567
206 29999983333327
207 1003389245896063
208 122473544024777
209 75498211766822
210 122057363546367
211 62441868958167
212 70695827316667
213 1019258504943167
214 134081680328813
215 89324067192437
216 1032947233308167
217 218837793765083
218 1033914890061063
219 244744764757083
220 314451904109293
221 10029346433286667
222 1048284312531667
223 599999983333333
224 10033941896856833
225 1182788226522383
226 608275330724417
227 1260515763483333
228 836594274358167
229 943291047332683
230 10098465179892437
231 1360146675177313
232 100099943251726117
233 10342571681597764
234 2167948082842133
235 10339143096407313
236 2445403011773313
237 2641018364192114
238 100337824323083167
239 12206133703183363
240 6927459779738887
241 100493283294402313
242 13036794084382433
243 5375861596060663
244 13034531061556833
245 9983486364492063
246 9949874270443813
247 102371381541913363
248 24474467305722917
249 1000449848766989417
250 108443524467807563
251 16733200200191833
252 109535381471001414
253 44698630849165614
254 31622774688331667
255 1004290788964973924
256 109357212336452683
257 70494609012957083
258 1008910252647671333
259 132663095727108637
260 83066231922477313
261 167032630943744043
262 141062751917357687
263 10006797119958013813
264 1043791118657318333
265 173117849451724613
266 10023761070361763167
267 1044020483515044833
268 299999998333333327
269 1086277587450785937
270 435866837461509417
271 447213595487659543
272 10049855670555642313
273 1180254209818799117
274 884872476574958167
275 10147304464727959417
276 1371087155507981917
277 758946572361980563
278 1996994987474680563
279 707106074079263583
280 100049937411274724684
281 10486658189331989417
282 2785480209927724417
283 100239159011830301264
284 10478539974576557867
285 5999999998333333333
286 12453874858814022867
287 5467172934890572764
288 7063993189247842133
289 100279608495445872313
290 13709449219384307563
291 9429681807356492126
292 100925219344309472687
293 13037982934483960583
294 102956243073875704626
295 14141782065920722917
296 22331368049897411063
297 1001798381911169778167
298 101979845998050016667
299 29964628614384477133
300 1003289040586001809417
301 122474487053377692437
302 69280435174650779614
303 126014280452653451637
304 77453069648658793167
305 89441594903993074417
306 1013403028803841694833
307 130268952475254048517
308 10004444011932396734687
309 1038214811924680151786
310 141378562727837168333
311 94180040294109027313
312 262087386170528775387
313 223584183022838178583
314 10003949119062925341833
315 1095445064756466913583
316 435877272864734253937
317 10033942246091004683363
318 1256940609965275041833
319 741551749981004224417
320 1303755689873988238887
321 754718284918279954614
323 10310188644733906962083
324 1413081381198439327133
326 10317926146760258672687
327 2190638491396718286667
328 10428805779666241294542
329 2827719752694560960583
330 2999999999833333333327
332 11704272510480903443167
333 2976388751488907738914
335 12609000708220259236417
336 8244385168100697671333
337 13408542049715882477313
338 8882505274864168010583
341 22360389084025298775583
344 24490814584000030724417
346 43566041821463294027313
347 44721359538546565724417
350 70497517615005252750707
353 83486465909152004768333
355 99689518004050952477133
但是现在这种方案找到的结果会遇上比较大的问题
比如263找到的当前最佳结果为10006797119958013813，长度20，其平方为100135988599999999889698999959898798969（长度39），数码和为283，搜索自由度需要(39*9-283=68)。
比较有利的条件是差值为263的数据，其长度如果小于20，那么由于平方和模9余数只能为0，1，4，7, 而283模9余数为4， 那么只能让寻找长度降低到17，数码和为280的数。
而长度为17的平方数长度为17*2=34,如果数字全部为9和才为34*9=306,给280的自由度只有26，远远小于我们的搜索范围41，所以不需要考虑，唯一的问题是20位数中是否还有小于
10006797119958013813的数，因为现在只使用了自由度41，而我们需要验证自由度小于68的数，比较有利的是我们只需要搜索小于10013或者更小开头的5位数，这5位数码和最大为10009，已经占用了35的自由度，对应搜索空间其实也不大，只是需要再修改专门的程序找出是否有更小的同样位数的结果 (所有需要验证的数据都应该是平方后前几位比较小的，特别对于整数比较小的时候）。
111 requires 50 with value 1039225192 and square 1003599887979998929
114 requires 47 with value 1029563014 and square 1006987999564998889
128 requires 50 with value 10344080383 and square 100578897669998896996
131 requires 47 with value 10392303883 and square 100599989558899888996
136 requires 59 with value 103435970528 and square 10008979987899889938496
145 requires 50 with value 104345579684 and square 10068984897696898998889
148 requires 47 with value 103923000293 and square 10099798993998799994896
153 requires 59 with value 1034407994942 and square 1000988298688985989999969
156 requires 56 with value 1039177559417 and square 1003583888989878989799889
162 requires 50 with value 1038749247412 and square 1008779599877899859889889
165 requires 47 with value 1043546836513 and square 1008989989798979887879489
170 requires 59 with value 10392299986043 and square 100096389888998998898978689
173 requires 56 with value 10440253828327 and square 100379999996978967985987969
179 requires 50 with value 10440306494543 and square 100989867879997999858788889
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Jack315 发表于 2024-2-25 11:24
Python 的整数类型有无限精度，正好用来做这个题。
附件是 Python 代码和计算结果 (n = 1 ~ 2048) :
谢谢 Jack315！ 宝贵资料我慢慢来消化。谢谢 Jack315！
主帖增加后变成如下的题目：
A*A各个数位上的数码和=9*1。A最小=3。
A*A各个数位上的数码和=9*2。A最小=24。
A*A各个数位上的数码和=9*3。A最小=63。
A*A各个数位上的数码和=9*4。A最小=264。
A*A各个数位上的数码和=9*5。A最小=1374。
......
T(01)=3^2=9,
T(02)=24^2=576,
T(03)=63^2=3969,
T(04)=264^2=69696,
T(05)=1374^2=1887876,
T(06)=3114^2=9696996,
T(07)=8937^2=79869969,
T(08)=60663^2=3679999569,
T(09)=94863^2=8998988769,
......
详见31#。欢迎 Jack315参与！谢谢！
northwolves 发表于 2024-2-24 13:54
3 9 9
707106074079263583 499998999999788997978888999589997889 33/4
943345110232670883 889899996999 ...
487/59,513/62很好：与62#,63#吻合。再来几个？谢谢！
我们是闹着玩的。享受的是解题过程。还有：
数字串T(n): x*x各个数位上的数码和=9*T。
数字串A(n): x*x各个数位上的数码和=9(x+A)。
数字串A(n)是数字串T(n)的子集。
若能证明：数字串A(n)中 x 的高位数都是>sqrt{10}的,则62#的可信度又提了一截。
789的找到一个：$9949370777987917^2 =98989978877879888789778997998889$
oeis搜索了一下，只有两项：
A058471 Numbers k such that k^2 contains only digits {7,8,9}.
3, 9949370777987917