这样的A有多少个？ - 第10页 - 开心茶馆 - 数学研发论坛

王守恩 发表于 2024-2-28 15:32:00

先立4根标杆(无限数串, 极限性价比(数码和: 数位)=33:4),慢慢再来找更好的性价比。
标配11位数。11位数性价比=088/11,15位数性价比=121/15,19位数性价比=154/15, ......
标配12位数。12位数性价比=082/12,16位数性价比=115/16,20位数性价比=148/16, ......
标配13位数。13位数性价比=094/13,17位数性价比=127/17,21位数性价比=160/17, ......
标配14位数。14位数性价比=106/14,18位数性价比=139/11,22位数性价比=172/11, ......
标配11位数。Table // TableForm
{"298327", "88998998929"},
{"29983327", "898999897988929"},
{"2999833327", "8998999989779888929"},
{"299998333327", "89998999998977798888929"},
{"29999983333327", "899998999999897777988888929"},
{"2999999833333327", "8999998999999989777779888888929"},
{"299999998333333327", "89999998999999998977777798888888929"},
{"29999999983333333327", "899999998999999999897777777988888888929"}
Table[(33 n + 55)/(4 n + 7), {n, 20}]
{8, 121/15, 154/19, 187/23, 220/27, 253/31, 286/35, 319/39, 352/43, 385/47, 418/51, 41/5, 484/59, 517/63, 550/67, 583/71, 616/75, 649/79, 682/83, 715/87}
标配12位数。Table // TableForm
{"58", "3364"},
{"5983", "35796289"},
{"599833", "359799627889"},
{"59998333", "3599799962778889"},
{"5999983333", "35999799996277788889"},
{"599999833333", "359999799999627777888889"},
{"59999998333333", "3599999799999962777778888889"},
{"5999999983333333", "35999999799999996277777788888889"},
{"599999999833333333", "359999999799999999627777777888888889"},
{"59999999998333333333", "3599999999799999999962777777778888888889"},
{"5999999999983333333333", "35999999999799999999996277777777788888888889"}
Table[(33 n - 17)/(4 n), {n, 20}]
{4, 49/8, 41/6, 115/16, 37/5, 181/24, 107/14, 247/32, 70/9, 313/40, 173/22, 379/48, 103/13, 445/56, 239/30, 511/64, 8, 577/72, 305/38, 643/80}
Table[标配13位数。k = 3 100^n - (5*10^(n - 1) + 1)/3; {k, k^2}, {n, 10}] // TableForm
{"298", "88804"},
{"29983", "898980289"},
{"2999833", "8998998027889"},
{"299998333", "89998999802778889"},
{"29999983333", "899998999980277788889"},
{"2999999833333", "8999998999998027777888889"},
{"299999998333333", "89999998999999802777778888889"},
{"29999999983333333", "899999998999999980277777788888889"},
{"2999999999833333333", "8999999998999999998027777777888888889"},
{"299999999998333333333", "89999999998999999999802777777778888888889"}
Table[(33 n - 5)/(4 n + 1), {n, 20}]
{28/5, 61/9, 94/13, 127/17, 160/21, 193/25, 226/29, 259/33, 292/37, 325/41, 358/45, 391/49, 8, 457/57, 490/61, 523/65, 556/69, 589/73, 622/77, 655/81}
标配14位数。Table // TableForm
{"583", "339889"},
{"59833", "3579987889"},
{"5998333", "35979998778889"},
{"599983333", "359979999877788889"},
{"59999833333", "3599979999987777888889"},
{"5999998333333", "35999979999998777778888889"},
{"599999983333333", "359999979999999877777788888889"},
{"59999999833333333", "3599999979999999987777777888888889"},
{"5999999998333333333", "35999999979999999998777777778888888889"},
{"599999999983333333333", "359999999979999999999877777777788888888889"}
Table[(33 n + 7)/(4 n + 2), {n, 20}]
{20/3, 73/10, 53/7, 139/18, 86/11, 205/26, 119/15, 271/34, 8, 337/42, 185/23, 403/50, 218/27, 469/58, 251/31, 535/66, 284/35, 601/74, 317/39, 667/82}

王守恩 发表于 2024-2-29 13:17:48

展望题目的困难程度。

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,
T(10)=545793^2=297889998849,
T(11)=1989417^2=3957779999889,
T(12)=5477133^2=29998985899689,
T(13)=20736417^2=429998989997889,
T(14)=82395387^2=6788999798879769,
T(15)=260191833^2=67699789959899889,
T(16)=706399164^2=498999778899898896,
T(17)=2428989417^2=5899989587897999889,
T(18)=9380293167^2=87989899898866889889,
T(19)=28105157886^2=789899899796987988996,
T(20)=99497231067^2=9899698989999989958489,
T(21)=538479339417^2=289959998978968689899889,
T(22)=1974763271886^2=3899689979989899957996996,
T(23)=4472135831667^2=19999998896879889759998889,
T(24)=14106593458167^2=198995978993999999978999889,
T(25)=62441868958167^2=3898986998988899589995999889,
T(26)=244744764757083^2=59899999875999896899998668889,
T(27)=836594274358167^2=699889979888867998799799599889,
T(28)=2445403011773313^2=5979995889989989998888898995969,
T(29)=9983486364492063^2=99669999989998948997699989995969,
T(30)=44698630849165614^2=1997967599789979898899879999996996,
T(31)=167032630943744043^2=27899899799988999988898998697985849,
T(32)=435866837461509417^2=189979899998697868879998999979679889,
T(33)=707106074079263583^2=499998999999788997978888999589997889,
T(34)=5467172934890572764^2=29889979899999998978989858999978599696,
T(35)=14141782065920722917^2=199989999999996989894777897899888988889,
T(36)=77453069648658793167^2=5998977997999989949998988998868885889889,
T(37)=262087386170528775387^2=68689797989699877986999999998989896999769,
T(38)=754718284918279954614^2=569599689589989999998989999797889899888996,
T(39)=2827719752694560960583^2=7995998999778988998899699998797979679699889,
T(40)=8882505274864168010583^2=78898899957989768899997956979998979999999889,
T(41)=43566041821463294027313^2=1897999999989488769979889989869689898789999969,
T(42)=99689518004050952477133^2=9937999999879998999788999799759989988887899689,
T(43)=315892386718941028010583^2=99787999986988989979987995998989889798759999889,
T(44)=893241282627485818275387^2=797879988989995997899989877988999997998969999769,
45-403=5999999999998333333333333^2=35999999999979999999999998777777777778888888888889,
46-418=29999999999983333333333327^2=899999999998999999999999897777777777988888888888929,
47-424=299999999999998333333333333^2=89999999999998999999999999802777777777778888888888889,
48-436=599999999999983333333333333^2=359999999999979999999999999877777777777788888888888889,
49-445=5999999999999983333333333333^2=35999999999999799999999999996277777777777788888888888889,
50-451=2999999999999833333333333327^2=8999999999998999999999999989777777777779888888888888929,
51-457=29999999999999983333333333333^2=899999999999998999999999999980277777777777788888888888889,
52-469=59999999999999833333333333333^2=3599999999999979999999999999987777777777777888888888888889,
53-478=599999999999999833333333333333^2=359999999999999799999999999999627777777777777888888888888889,
54-484=299999999999998333333333333327^2=89999999999998999999999999998977777777777798888888888888929,
55-490=2999999999999999833333333333333^2=8999999999999998999999999999998027777777777777888888888888889,
56-502=5999999999999998333333333333333^2=35999999999999979999999999999998777777777777778888888888888889,
57-513=9984988582817657883693383344833^2=99699996998998979989989997788978889798779999999969798987797889,
57-517=29999999999999983333333333333327^2=899999999999998999999999999999897777777777777988888888888888929,
58-523=299999999999999998333333333333333^2=89999999999999998999999999999999802777777777777778888888888888889,
59-535=599999999999999983333333333333333^2=359999999999999979999999999999999877777777777777788888888888888889,
60-544=5999999999999999983333333333333333^2=35999999999999999799999999999999996277777777777777788888888888888889,
61-550=2999999999999999833333333333333327^2=8999999999999998999999999999999989777777777777779888888888888888929,
62-556=29999999999999999983333333333333333^2=899999999999999998999999999999999980277777777777777788888888888888889,
63-568=59999999999999999833333333333333333^2=3599999999999999979999999999999999987777777777777777888888888888888889,
64-577=599999999999999999833333333333333333^2=359999999999999999799999999999999999627777777777777777888888888888888889,
65-583=299999999999999998333333333333333327^2=89999999999999998999999999999999998977777777777777798888888888888888929,
66-589=2999999999999999999833333333333333333^2=8999999999999999998999999999999999998027777777777777777888888888888888889,,
67-601=5999999999999999998333333333333333333^2=35999999999999999979999999999999999998777777777777777778888888888888888889,
68-616=29999999999999999983333333333333333327^2=899999999999999998999999999999999999897777777777777777988888888888888888929,,
69-622=299999999999999999998333333333333333333^2=89999999999999999998999999999999999999802777777777777777778888888888888888889,,
70-634=599999999999999999983333333333333333333^2=359999999999999999979999999999999999999877777777777777777788888888888888888889,
71-643=5999999999999999999983333333333333333333^2=35999999999999999999799999999999999999996277777777777777777788888888888888888889,
72-649=2999999999999999999833333333333333333327^2=8999999999999999998999999999999999999989777777777777777779888888888888888888929,
73-655=29999999999999999999983333333333333333333^2=899999999999999999998999999999999999999980277777777777777777788888888888888888889,
74-557=59999999999999999999833333333333333333333^2=3599999999999999999979999999999999999999987777777777777777777888888888888888888889,
75-676=599999999999999999999833333333333333333333^2=359999999999999999999799999999999999999999627777777777777777777888888888888888888889,
76-682=299999999999999999998333333333333333333327^2=89999999999999999998999999999999999999998977777777777777777798888888888888888888929,
77-688=2999999999999999999999833333333333333333333^2=8999999999999999999998999999999999999999998027777777777777777777888888888888888888889,
78-700=5999999999999999999998333333333333333333333^2=35999999999999999999979999999999999999999998777777777777777777778888888888888888888889,
79-715=29999999999999999999983333333333333333333327^2=899999999999999999998999999999999999999999897777777777777777777988888888888888888888929,
80-721=299999999999999999999998333333333333333333333^2=89999999999999999999998999999999999999999999802777777777777777777778888888888888888888889,
81-733=599999999999999999999983333333333333333333333^2=359999999999999999999979999999999999999999999877777777777777777777788888888888888888888889,
82-742=5999999999999999999999983333333333333333333333^2=35999999999999999999999799999999999999999999996277777777777777777777788888888888888888888889,
83-748=2999999999999999999999833333333333333333333327^2=8999999999999999999998999999999999999999999989777777777777777777779888888888888888888888929,
84-754=29999999999999999999999983333333333333333333333^2=899999999999999999999998999999999999999999999980277777777777777777777788888888888888888888889,
85-766=59999999999999999999999833333333333333333333333^2=3599999999999999999999979999999999999999999999987777777777777777777777888888888888888888888889,
86-775=599999999999999999999999833333333333333333333333^2=359999999999999999999999799999999999999999999999627777777777777777777777888888888888888888888889,
87-781=299999999999999999999998333333333333333333333327^2=89999999999999999999998999999999999999999999998977777777777777777777798888888888888888888888929,
88-787=2999999999999999999999999833333333333333333333333^2=8999999999999999999999998999999999999999999999998027777777777777777777777888888888888888888888889,
89-799=5999999999999999999999998333333333333333333333333^2=35999999999999999999999979999999999999999999999998777777777777777777777778888888888888888888888889,
90-814=29999999999999999999999983333333333333333333333327^2=899999999999999999999998999999999999999999999999897777777777777777777777988888888888888888888888929,
91-820=299999999999999999999999998333333333333333333333333^2=89999999999999999999999998999999999999999999999999802777777777777777777777778888888888888888888888889,
92-832=599999999999999999999999983333333333333333333333333^2=359999999999999999999999979999999999999999999999999877777777777777777777777788888888888888888888888889,
93-841=5999999999999999999999999983333333333333333333333333^2=35999999999999999999999999799999999999999999999999996277777777777777777777777788888888888888888888888889,
94-847=2999999999999999999999999833333333333333333333333327^2=8999999999999999999999998999999999999999999999999989777777777777777777777779888888888888888888888888929,
95-853=29999999999999999999999999983333333333333333333333333^2=899999999999999999999999998999999999999999999999999980277777777777777777777777788888888888888888888888889,
96-865=59999999999999999999999999833333333333333333333333333^2=3599999999999999999999999979999999999999999999999999987777777777777777777777777888888888888888888888888889,
97-874=599999999999999999999999999833333333333333333333333333^2=359999999999999999999999999799999999999999999999999999627777777777777777777777777888888888888888888888888889,
98-880=299999999999999999999999998333333333333333333333333327^2=89999999999999999999999998999999999999999999999999998977777777777777777777777798888888888888888888888888929,
99-886=2999999999999999999999999999833333333333333333333333333^2=8999999999999999999999999998999999999999999999999999998027777777777777777777777777888888888888888888888888889,

王守恩 发表于 2024-2-29 13:19:43

接31楼。谢谢 mathe 版主！！！

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,
T(10)=545793^2=297889998849,
T(11)=1989417^2=3957779999889,
T(12)=5477133^2=29998985899689,
T(13)=20736417^2=429998989997889,
T(14)=82395387^2=6788999798879769,
T(15)=260191833^2=67699789959899889,
T(16)=706399164^2=498999778899898896,
T(17)=2428989417^2=5899989587897999889,
T(18)=9380293167^2=87989899898866889889,
T(19)=28105157886^2=789899899796987988996,
T(20)=99497231067^2=9899698989999989958489,
T(21)=538479339417^2=289959998978968689899889,
T(22)=1974763271886^2=3899689979989899957996996,
T(23)=4472135831667^2=19999998896879889759998889,
T(24)=14106593458167^2=198995978993999999978999889,
T(25)=62441868958167^2=3898986998988899589995999889,
T(26)=244744764757083^2=59899999875999896899998668889,
T(27)=836594274358167^2=699889979888867998799799599889,
T(28)=2445403011773313^2=5979995889989989998888898995969,
T(29)=9983486364492063^2=99669999989998948997699989995969,
T(30)=44698630849165614^2=1997967599789979898899879999996996,
T(31)=167032630943744043^2=27899899799988999988898998697985849,
T(32)=435866837461509417^2=189979899998697868879998999979679889,
T(33)=707106074079263583^2=499998999999788997978888999589997889,
T(34)=5467172934890572764^2=29889979899999998978989858999978599696,
T(35)=14141782065920722917^2=199989999999996989894777897899888988889,
T(36)=77453069648658793167^2=5998977997999989949998988998868885889889,
T(37)=262087386170528775387^2=68689797989699877986999999998989896999769,
T(38)=754718284918279954614^2=569599689589989999998989999797889899888996,
T(39)=2827719752694560960583^2=7995998999778988998899699998797979679699889,
T(40)=8882505274864168010583^2=78898899957989768899997956979998979999999889,
T(41)=43566041821463294027313^2=1897999999989488769979889989869689898789999969,
T(42)=99689518004050952477133^2=9937999999879998999788999799759989988887899689,
T(43)=315892386718941028010583^2=99787999986988989979987995998989889798759999889,
T(44)=893241282627485818275387^2=797879988989995997899989877988999997998969999769,

王守恩 发表于 2024-2-29 18:33:28

northwolves 发表于 2024-2-24 13:54
3 9 9
707106074079263583 499998999999788997978888999589997889 33/4
943345110232670883 889899996999 ...
这串数能再来几个？ {3, 83, 313, 94863, 298327, 987917} 谢谢！
Select, Total] >= 11*(3*IntegerLength[#^2] - 1)/4 &]
也就是91#标配11位数(我把它4等分了)。11位数性价比=088/11,15位数性价比=121/15,19位数性价比=154/19, ......

王守恩 发表于 2024-3-1 10:28:31

先叉一叉。为找一串极限性价比>33/4的无限数串垫垫底！

x由数码0, 5组成(个位数=5)，x*x由数码0, 2, 5组成。

x=5, 505, 5005, 50005, 500005, 500505, 5000005, 5000505, 5050005, 50000005, 50000505, 50005005, 50500005, 50000005, ......

查了。OEIS没有这串数。

northwolves 发表于 2024-3-1 10:44:18

本帖最后由 northwolves 于 2024-3-1 11:24 编辑

王守恩发表于 2024-2-29 18:33
这串数能再来几个？ {3, 83, 313, 94863, 298327, 987917} 谢谢！

也就是91#标配11位数(我把它4等分了) ...

{3,83,313,94863,298327,987917,3162083,29983327,99477133,99483667,994927133,2983284917,2999833327,28105157886,3160522105583}

northwolves 发表于 2024-3-1 11:03:28

王守恩发表于 2024-3-1 10:28
先叉一叉。为找一串极限性价比>33/4的无限数串垫垫底！

x由数码0, 5组成(个位数=5)，x*x由数码0, 2, 5组成 ...

$5*\ 10^{n+1}+5,5*\ 10^{n+4}+505,505*\ 10^{n+3}+5$

nyy 发表于 2024-3-1 12:21:31

老同志玩这东西很开心

northwolves 发表于 2024-3-2 21:01:34

能再来2个？谢谢！
---------------------------
94867796958167

王守恩 发表于 2024-3-5 13:04:31

本帖最后由王守恩于 2024-3-5 14:05 编辑

王守恩发表于 2024-2-28 15:32
先立4根标杆(无限数串, 极限性价比(数码和: 数位)=33:4),慢慢再来找更好的性价比。
标配11位数。11位数性 ...
小结。合并 91#(标配13位数有提升),92#。就无限数串, 好像不会有更好的性价比>33/4。
FullSimplify@Table - 4) Cos] - 8 - Sqrt) - 5*10^(n/4) (2 (11 Sqrt - 20) (10 Cos[(n*Pi/2] - 10^(1/4) Sin[(n*Pi/2]) +
9 Sqrt ((10^(1/4) - 10) Cos - (10^(1/4) - (10^(1/4) - 10) I Sin + 10))) - 4 (10 + 9 Cos) (Sqrt - 10))/(12 (Sqrt - 10)) ; {k, k^2}, {n, 40}] // TableForm
{"583", "339889"},
{"2827", "7991929"},
{"5983", "35796289"},
{"29827", "889649929"},
{"59833", "3579987889"},
{"298327", "88998998929"},
{"599833", "359799627889"},
{"2998327", "8989964798929"},
{"5998333", "35979998778889"},
{"29983327", "898999897988929"},
{"59998333", "3599799962778889"},
{"299983327", "89989996477988929"},
{"599983333", "359979999877788889"},
{"2999833327", "8998999989779888929"},
{"5999983333", "35999799996277788889"},
{"29999833327", "899989999647779888929"},
{"59999833333", "3599979999987777888889"},
{"299998333327", "89998999998977798888929"},
{"599999833333", "359999799999627777888889"},
{"2999998333327", "8999989999964777798888929"},
{"5999998333333", "35999979999998777778888889"},
{"29999983333327", "899998999999897777988888929"},
{"59999998333333", "3599999799999962777778888889"},
{"299999983333327", "89999989999996477777988888929"},
{"599999983333333", "359999979999999877777788888889"},
{"2999999833333327", "8999998999999989777779888888929"},
{"5999999983333333", "35999999799999996277777788888889"},
{"29999999833333327", "899999989999999647777779888888929"},
{"59999999833333333", "3599999979999999987777777888888889"},
{"299999998333333327", "89999998999999998977777798888888929"},
{"599999999833333333", "359999999799999999627777777888888889"},
{"2999999998333333327", "8999999989999999964777777798888888929"},
{"5999999998333333333", "35999999979999999998777777778888888889"},
{"29999999983333333327", "899999998999999999897777777988888888929"},
{"59999999998333333333", "3599999999799999999962777777778888888889"},
{"299999999983333333327", "89999999989999999996477777777988888888929"},
{"599999999983333333333", "359999999979999999999877777777788888888889"},
{"2999999999833333333327", "8999999998999999999989777777779888888888929"},
{"5999999999983333333333", "35999999999799999999996277777777788888888889"},
{"29999999999833333333327", "899999999989999999999647777777779888888888929"}
Table[( 66 n + 27 Cos - 33 Cos + 15 ((1 - I) (-I)^n + (1 + I) I^n) + 152)/(8 (n + 3)), {n, 3, 54}]
{20/3, 55/7, 49/8, 64/9, 73/10, 8, 41/6, 97/13, 53/7, 121/15, 115/16, 130/17, 139/18, 154/19, 37/5, 163/21, 86/11, 187/23, 181/24, 196/25, 205/26, 220/27, 107/14, 229/29, 119/15,253/31, 247/32,
262/33, 271/34, 286/35, 70/9, 295/37, 8, 319/39, 313/40, 8, 337/42, 352/43, 173/22, 361/45, 185/23, 385/47, 379/48, 394/49, 403/50, 418/51, 103/13, 427/53, 218/27, 41/5, 445/56, 460/57}

补充内容 (2024-3-6 09:09):
通俗一点: 40/6, 55/7, 49/8, 64/9, 73/10, 88/11, 82/12, 97/13, 106/14,...=数码和/数位的最大值。

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