数字串的通项公式

王守恩

备忘。{2, 5, 23, 110, 527, 2525, 12098, 57965, 277727, 1330670, 6375623, 30547445, 146361602, 701260565, 3359941223, 16098445550, 77132286527, 369562987085, 1770682648898, 8483850257405}

1. Table[LucasL[n, 5 I]/I^n, {n, 0, 19}]

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northwolves

王守恩 发表于 2024-12-3 14:57
备忘。{2, 5, 23, 110, 527, 2525, 12098, 57965, 277727, 1330670, 6375623, 30547445, 146361602, 701260 ...

$a_n=\lfloor \left(\frac{5+\sqrt{21}}{2} \right)^{n-1}+1\rfloor$

王守恩

9*9*9*9*9*9*......,  积的最高位数字=9。

9^22=984770902183611232881。
9^44=969773729787523602876821942164080815560161。
9^66=955004950796825236893190701774414011919935138974343129836853841。
9^88=940461086986004843694934910131056317906479029659199959555574885740211572136210345921。
9^110=926138713099787670959935798024513966701772293499227988263405269197039529170894882252068039219702299428401。
9^132=912034456046446591670769941126967809732389880154759674362919253085466672523897586208912607420113148072606337611541329196453281。
9^153=99793888233710926097676673961542382339552034110870991187709058567130998942396826836880350287497238272034603157195937657211050782186192219658614729。
9^175=98274117348321974353044780928022697503543794108996224149902690255438168118107927224939057895356483251830948245334782867413814443266637838233302304694183773324275704249。

22, 44, 66, 88, 110, 132, 153, 175, 197, 219, 241, 263, 285, 306, 328, 350, 372, 394, 416, 438, 459, 481, 503, 525, 547, 569, 591, 612, 634, 656, 678, 700, 722, 744, 765, 787, 809, 831, 853, 875, 897, 918, 940, 962, 984, 1006,
1028, 1050, 1071, 1093, 1115, 1137, 1159, 1181, 1202, 1224, 1246, 1268, 1290, 1312, 1334, 1355, 1377, 1399, 1421, 1443, 1465, 1487, 1508, 1530, 1552, 1574, 1596, 1618, 1640, 1661, 1683, 1705, 1727, 1749, 1771, 1793,
1814, 1836, 1858, 1880, 1902, 1924, 1946, 1967, 1989, 2011, 2033, 2055, 2077, 2099, 2120, 2142, 2164, 2186, 2208, 2230, 2251, 2273, 2295, 2317, 2339, 2361, 2383, 2404, 2426, 2448, 2470, 2492, 2514, 2536, 2557, 2579,
2601, 2623, 2645, 2667, 2689, 2710, 2732, 2754, 2776, 2798, 2820, 2842, 2863, 2885, 2907, 2929, 2951, 2973, 2995, 3016, 3038, 3060, 3082, 3104, 3126, 3148, 3169, 3191, 3213, 3235, 3257, 3279, 3301, 3322, 3344, 3366,
3388, 3410, 3432, 3453, 3475, 3497, 3519, 3541, 3563, 3585, 3606, 3628, 3650, 3672, 3694, 3716, 3738, 3759, 3781, 3803, 3825, 3847, 3869, 3891, 3912, 3934, 3956, 3978, 4000, 4022, 4044, 4065, 4087, 4109, 4131,  ......

公式(1)。Flatten@Table[Solve[{9*10^(x-a) < 9^x <10*10^(x-a), x > 0}, {x}, Integers], {a, 189}]

公式(2)。Select[Range[4131], Length[IntegerDigits[9^# ]] == Length[IntegerDigits[9^#*9]] &]

公式(3)。Select[Range[4131], IntegerLength[9^(#)] == IntegerLength[9^(# + 1)] &]

公式(4)。Table[Ceiling[(n*Log[10])/(Log[10] - Log[9])], {n, 189}]

求助。怎么把公式(1)答案中的"x"去掉。谢谢！这些按钮我怎么也学不好了。

重点在于公式(1)可以变化。譬如：

公式(1)。Flatten@Table[Solve[{9*10^(x+a) < 18^x <10*10^(x+a), x > 0}, {x}, Integers], {a, 189}]。还是那么好运,可以有公式(2),公式(3),公式(4)吗？

王守恩

northwolves 发表于 2024-12-3 15:35
$a_n=\lfloor \left(\frac{5+\sqrt{21}}{2} \right)^{n-1}+1\rfloor$

接楼上。求助。怎么把公式(1)答案中的"x"去掉。谢谢！这些按钮我怎么也学不好了。

重点在于公式(1)可以变化,可以掌控,速度也不慢。譬如：

公式(1)。Flatten@Table[Solve[{9*10^(x+a) < 11^x <10*10^(x+a), x > 0}, {x}, Integers], {a, 189}]。不敢贪心：还是那么好运,可以有公式(2),公式(3),公式(4)吗？

{x -> 48, x -> 72, x -> 96, x -> 120, x -> 144, x -> 169, x -> 193, x -> 217, x -> 241, x -> 265, x -> 289, x -> 313, x -> 314, x -> 338, x -> 362, x -> 386, x -> 410, x -> 434, x -> 458, x -> 459,

northwolves

这个代码更直观些：

1. Select[Range@4131, First@IntegerDigits[9^#] == 9 &]

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王守恩

northwolves 发表于 2024-12-9 09:19
这个代码更直观些：

1. Flatten@Table[Solve[{5*10^(x + a) < 17^x < 6*10^(x + a), x > 0}, {x}, Integers], {a, 49}]

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{x -> 12, x -> 25, x -> 38, x -> 51, x -> 64, x -> 77, x -> 90, x -> 103, x -> 116, x -> 129, x -> 142, x -> 155, x -> 168, x -> 181,x -> 194, x -> 207, x -> 216}

1. Select[Range@4131, First@IntegerDigits[17^#] == 5 &]

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{12, 25, 38, 51, 64, 77, 90, 103, 116, 129, 142, 155, 168, 181, 194, 207, 216, 229, 242, 255, 268, 281, 294, 307, 320, 333, 346, 359, 372, 385, 398, 411, 424, 437, 450, 459, 472, 485, 498, 511, 524, 537, 550, 563, 576, 589, 602, 615,

k*k*k*k*k*k*......, 积的最高位数字=a。 这里k=17, a=5。

王守恩

northwolves 发表于 2024-12-9 09:19
这个代码更直观些：

6*6*6*6*6*6*......, 积的最高位数字=9。 会有吗？

6*6*6*6*6*6*......, 积的最高位数字=666。 怎么编排？

这个帖子是你的。查看: 804774|回复: 740。我才不管是怎么来的。反正这个帖子是属于你的。

northwolves

王守恩 发表于 2024-12-9 15:32
6*6*6*6*6*6*......, 积的最高位数字=9。 会有吗？

1. Select[Range@10000, First@IntegerDigits[6^#] == 9 &]

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{176,185,194,203,212,221,230,239,248,257,266,275,284,293,478,487,496,505,514,523,532,541,550,559,568,577,586,595,771,780,789,798,807,816,825,834,843,852,861,870,879,888,1073,1082,1091,1100,1109,1118,1127,1136,1145,1154,1163,1172,1181,1190,1366,1375,1384,1393,1402,1411,1420,1429,1438,1447,1456,1465,1474,1483,1668,1677,1686,1695,1704,1713,1722,1731,1740,1749,1758,1767,1776,1785,1961,1970,1979,1988,1997,2006,2015,2024,2033,2042,2051,2060,2069,2078,2263,2272,2281,2290,2299,2308,2317,2326,2335,2344,2353,2362,2371,2380,2556,2565,2574,2583,2592,2601,2610,2619,2628,2637,2646,2655,2664,2673,2858,2867,2876,2885,2894,2903,2912,2921,2930,2939,2948,2957,2966,2975,3151,3160,3169,3178,3187,3196,3205,3214,3223,3232,3241,3250,3259,3268,3453,3462,3471,3480,3489,3498,3507,3516,3525,3534,3543,3552,3561,3570,3746,3755,3764,3773,3782,3791,3800,3809,3818,3827,3836,3845,3854,3863,4048,4057,4066,4075,4084,4093,4102,4111,4120,4129,4138,4147,4156,4165,4341,4350,4359,4368,4377,4386,4395,4404,4413,4422,4431,4440,4449,4458,4643,4652,4661,4670,4679,4688,4697,4706,4715,4724,4733,4742,4751,4760,4936,4945,4954,4963,4972,4981,4990,4999,5008,5017,5026,5035,5044,5053,5238,5247,5256,5265,5274,5283,5292,5301,5310,5319,5328,5337,5346,5355,5531,5540,5549,5558,5567,5576,5585,5594,5603,5612,5621,5630,5639,5648,5833,5842,5851,5860,5869,5878,5887,5896,5905,5914,5923,5932,5941,5950,6126,6135,6144,6153,6162,6171,6180,6189,6198,6207,6216,6225,6234,6243,6428,6437,6446,6455,6464,6473,6482,6491,6500,6509,6518,6527,6536,6545,6721,6730,6739,6748,6757,6766,6775,6784,6793,6802,6811,6820,6829,6838,7023,7032,7041,7050,7059,7068,7077,7086,7095,7104,7113,7122,7131,7140,7316,7325,7334,7343,7352,7361,7370,7379,7388,7397,7406,7415,7424,7433,7618,7627,7636,7645,7654,7663,7672,7681,7690,7699,7708,7717,7726,7735,7911,7920,7929,7938,7947,7956,7965,7974,7983,7992,8001,8010,8019,8028,8213,8222,8231,8240,8249,8258,8267,8276,8285,8294,8303,8312,8321,8330,8506,8515,8524,8533,8542,8551,8560,8569,8578,8587,8596,8605,8614,8623,8808,8817,8826,8835,8844,8853,8862,8871,8880,8889,8898,8907,8916,8925,9101,9110,9119,9128,9137,9146,9155,9164,9173,9182,9191,9200,9209,9218,9403,9412,9421,9430,9439,9448,9457,9466,9475,9484,9493,9502,9511,9520,9696,9705,9714,9723,9732,9741,9750,9759,9768,9777,9786,9795,9804,9813,9998}

northwolves

王守恩 发表于 2024-12-9 17:12
6*6*6*6*6*6*......, 积的最高位数字=9。 会有吗？

6*6*6*6*6*6*......, 积的最高位数字=666。 怎么编排 ...

1. s = Select[Range[3, 100000], Take[IntegerDigits[6^#], 3] == {6, 6, 6} &]

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{420, 1015, 1610, 2205, 2800, 3395, 3990, 4585, 5180}

northwolves

6^5775=665994995174042149412776884305998414349235780237902046769212554197117295475990786172752471229554686079002142998067040831331921581670358640894191430839833944059372111644423260379654576444640628304623745199171610265167917196744087031392051571658641616386479536773946513062326713086964190933011443258709240301113130837448559138967658571567278836184013800308400401837559557728467702480700008845364998012232055209156946034590731898211018572308686318857554384897599280029430863766982472485615399785523391886765302057417804253369619390624210558195801834442308702325338167185557691760590004009886815060666832724748363317543287898612923715517451077426621419646918336693614238318862920336159871902482782566552313635402987347409153041071420209416777193232551382251058412019124118357365861620281023911952404698672342090519274763194969032479071290643209767583734582010329017968956213249253082193057864763912277906782507006391250110904070513095077206277676201948396600325131781337992392871573346688957991208212289331966883566840446143713124802473597779848377221178943527777219792361039519399764552336051469343381669436675754053381106363020168508608369742263632966984203269057959680519453042081603494680212490578686518469571742332345320099841047176295187879606531886630901617027065638783385961445506367926330306882786104178410625166387628937839505555049890145178941456247057558067300535751781693777475376363899877480034024148965050978044647926290674064177371988651343462218852122813280728461357914203735850139780532751518894583917162220918832441055650450714283187725950546604987165644839586105701269559618813953731796072894826419870215357719730739156110561122174187170782041044082254378181967676387146461558220611697588954584585691603386930522205372790097226505585542695058967561268423279326158505074276332115379649900861540285416020796947854749985273654373587267167152876710933489149082971298616805728173626644418003084784467614872198809483901568261089393306815013760777397915997654112937711417314706970754889305729030736050739992032577397161464829045995353743457305267253128894961734703402286112953253899294245337446722109349939891626193315103075212694741029887211069040791432047211514494615722055754728023417289600456822629558475548480922459536744998882206950044731891213884966834421306572592850336389558840851054639983087481693697484138339846873841229931177516866798747290665377906497583923978192220082144334334124361193292763222679204299219644679551152957033693787921275400762111969458641251109522517078513274750454342956837514054323958587479546033460823031357208504421336606221996032155856693379782248533264257850861394955027128284683626864513451843284050408657010730150915547523997106715619991533781670458681771816306327401133440273327452534154314097475858203581277528233057848235582468577943796603094825747748773092416070107830887693142112403510263880770337197486428499546918807411855976643414780043999283915739958821319441084761820835243131199120918579048606487663671074922654269517252495224323476487489796413484882729119959956118767699100083526937880170172116873539415623868045441814642657084327243450217902187288583950390171179541276567532347058840418177206685884922223541241821198430299426236500365737698843752268811320138382551173840470658909677725849045350123355340062336766077730161982658805033975452079491948580213908078590499574129227448852105780703448635674841452373885476886385836022541077078728418898311786785606903710525970499973320626881907194248751865634601927059608764904251388997833124824935771999859158097947218429536546618810923592889395011186287218840519627855503964380598266421777028205812723641208498285634276046612758056373468900686498460837195218588131626423546432243519980273202420860353497120356249077123537220220903466928180465916916508071583627722331706976060416747849717938467444152432371224347677745227157045680260320518320115186219747706662858764016443674284144758580494051753448741857622204993524366005265104409394879954485994147131961776304178171366743894386696530314028739912286595090658219433588647824061784341945303339319667197172150829160346211461480691876726207098910358923276139907035652859163150508759448981677136570784434602053919210178760376956909013929878646394975287462060273565505094347822422139911511019259310355159791292172968653041114743674738412963606272054520885232791639883305675422595081079010218038748509479424959794813362327517571122488386633253557818222332789470031237405448140746536461470085276033182151857924989318495552208041054759458885861376

差一点