以10为底的φ因子结构形式

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以10为底的φ因子结构形式

只看总的φ因子（素数或合数），其分解式不必关心。
以10为底的复合φ因子，都是以1结尾的整数，Φ1(10) =9除外。
类型一：特例，只有φ1一个，等于3的平方，以下各种类型的φ因子皆无平方因子。
Φ1(10) = 9=3^2

类型二：素数型，皆为清一色数。
Φ2(10)=11=11
Φ3(10)=111=3×37
Φ5(10)=11111=41×271
Φ7(10)=1111111=239×4649
Φ11(10)=11111111111<11>=21649×513239
Φ13(10)=1111111111111<13>=53×79×265371653
Φ17(10)=11111111111111111<17>=2071723×5363222357<10>
Φ19(10)=1111111111111111111<19>=1111111111111111111<p19>
Φ23(10)=11111111111111111111111<23>=11111111111111111111111<p23>
Φ29(10)=11111111111111111111111111111<29>=3191×16763×43037×62003×77843839397<11>
Φ31(10)=1111111111111111111111111111111<31>=2791×6943319×57336415063790604359<20>

类型三：2p型，p为奇素数，9090…91型（k组90加一组91型）。
Φ6(10)=91=7×13
Φ10(10)=9091=9091p
Φ14(10)=909091=909091p
Φ22(10)=9090909091<10>=11×23×4093×8779
Φ26(10)=909090909091<12>=859×1058313049<10>
Φ34(10)=9090909090909091<16>=103×4013×21993833369<11>
Φ38(10)=909090909090909091<18>=909090909090909091<p18>
Φ46(10)=9090909090909090909091<22>=47×139×2531×549797184491917<15>
Φ58(10)=9090909090909090909090909091<28>=59×154083204930662557781201849<27>
Φ62(10)=909090909090909090909090909091<30>=909090909090909090909090909091<p30>

类型四：3p型，p≥5，前半部k组900，后半部k-1组990加一组991；中间有一组90或没有。
Φ15(10)=900 90 991=31×2906161
Φ21(10)=900900 990991<12>=43×1933×10838689
Φ33(10)=900900900 90 990990991<20>=67×1344628210313298373<19>
Φ39(10)=900900900900 990990990991<24>=900900900900990990990991<p24>
Φ51(10)=900900900900900 90 990990990990991<32>=613×210631×52986961×13168164561429877<17>
Φ57(10)=900900900900900900 990990990990990991<36>=21319×10749631×3931123022305129377976519<25>
Φ69(10)=900900900900900900900 90 99099099099099099&#172;0991<44>
=277×203864078068831<15>×1595352086329224644348978893<28>
Φ87(10)=900900900900900900900900900 90 990990990990990990990990991<56>
=4003×72559×310170251658029759045157793237339498342763245483<48>
Φ93(10)=900900900900900900900900900900 990990990990990990990990990991<60>
=900900900900900900900900900900990990990990990990990990990991<p60>

类型五：4p型，p为奇素数，99009900…9901型（k组9900加一组9901型）。
Φ12(10)=9901=9901
Φ20L(10)=3541=3541
Φ20M(10)=27961=27961
Φ20(10)=99009901=3541×27961
Φ28(10)=990099009901<12>=29×281×121499449
Φ44(10)=99009900990099009901<20>=89×1052788969<10>×1056689261<10>
Φ52(10)=990099009900990099009901<24>=521×1900381976777332243781<22>
Φ68(10)=99009900990099009900990099009901<32>
=28559389×1491383821<10>×2324557465671829<16>
Φ76(10)=990099009900990099009900990099009901<36>
=722817036322379041<18>×1369778187490592461<19>
Φ92(10)=99009900990099009900990099009900990099009901<44>
=1289×18371524594609<14>×4181003300071669867932658901<28>

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类型六：5p型，p≥7，结构复杂，无统一规律。
Φ35(10)=900009090090909909099991<24>=71×123551×102598800232111471<18>
Φ55(10)=9000090000990009900099900999009999099991<40>
=1321×62921×83251631×1300635692678058358830121<25>
Φ65(10)=900009000090090900909009099090990909909999099991<48>
=162503518711<12>×5538396997364024056286510640780600481<37>
Φ85(10)=9000090000900009090090900909009090990909909099090999909999099991<64>
=262533041×8119594779271<13>×4222100119405530170179331190291488789678&#172;081<43>
Φ95(10)=900009000090000900099000990009900099009990099900999009990999909999099991<72>
=191×59281×63841×1289981231950849543985493631<28>×965194617121640791456070347951751<33>

类型七：6p型，内有多个6位小循环节109891，其中109+891=999；尾部两位是11，中部有不完全循环节。
Φ30(10)=109889011=211×241×2161
Φ42(10)=1098900989011<13>=7×127×2689×459691
Φ66(10)=109890109889010989011<21>=599144041×183411838171<12>
Φ78(10)=1098901098900989010989011<25>=13×157×6397×216451×388847808493<12>
Φ102(10)=109890109890109889010989010989011<33>
=291078844423<12>×377526955309799110357<21>
Φ114(10)=1098901098901098900989010989010989011<37>
=1458973×753201806271328462547977919407<30>
Φ138(10)=109890109890109890109889010989010989010989011<45>
=31051×143574021480139<15>×24649445347649059192745899<26>

类型八：7p型，p≥11，结构复杂，无统一规律。
Φ77(10)=900000090009009000900990090099009909900990999099099909999991<60>
=5237×42043×29920507×1366146685760023293714964475559157409101&#172;81043<45>
Φ91(10)=900000090000099000009900009990000999000999900099990099999009999909999991<72>
=547×14197×17837×4262077×43442141653<11>×316877365766624209<18>×110742186470530054291318013<27>
Φ119(10)=9000000900...<96>
=923441×3924966376871<13>×768736559421401249042753476963<30>
×3230129421485627516508145444373504546404&#172;48842187<48>
Φ161(10)=9000000900...<132>
=6763×472341157×11273170771131750391<20>×1626777403161656797092007877<28>
×15362898429170396757717888856328974146292496901433891193564055671816191643<74>

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类型九：8p型，p为奇素数，9999000099990000…99990001型（k组99990000加一组99990001型）。
Φ24(10)=99990001=99990001
Φ40(10)=9999000099990001<16>=1676321×5964848081<10>
Φ56(10)=999900009999000099990001<24>=7841×127522001020150503761<21>
Φ88(10)=9999000099990000999900009999000099990001<40>
=617×16205834846012967584927082656402106953<38>
Φ104(10)= 999900009999000099990000999900009999000099990001<48>
=1580801×632527440202150745090622412245443923049201<42>
Φ152(10)= 999900009999000099990000999900009999000099990000999900009999000099990001<72>
=457×1403417×5240808656722481737<19>×297478330786365628414805305290302483555043017<45>
Φ184(10)= 9999000099990000999900009999000099990000999900009999000099990000999900009999000099990001<88>
=2393×4178437150016715837818641871709193476807772628503969494400330129962348520684914375257<85>

类型十：10p型，p≥7
Φ70(10)=1099988890111109888900011<25>=4147571×265212793249617641<18>
Φ110(10)= 10999890000989990100100098999009998900011<41>
=331×5171×20163494891<11>×318727841165674579776721<24>
Φ130(10)= 1099989000109888901110988901110988890109998900011<49>
=131×8396862596258693901610602298557167100076327481<46>
Φ170(10)= 10999890001099988890111098889011110988890111098889000109998900011<65>
=87211×787223761×160220794821014452066741918303580917664386555934641<51>
Φ190(10)= 1099989000109998899901000989990100099010009899901000989989000109998900011.<73>
=1812604116731<13>×121450506296081<15>×4996731930447843676185843959746621491531100801<46>

类型十一：30倍数
Φ30(10)=109889011=211×241×2161
Φ60L(10)=255522961=61×4188901
Φ60M(10)=39526741=39526741p
Φ60(10)=10099989899000101<17>
Φ90(10)=1000999998998999000001001<25>=29611×3762091×8985695684401<13>
Φ150(10)= 10000099999999989999899999000000000100001<41>
=10000099999999989999899999000000000100001<p41>

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类型十二：p平方型，(p-1)小节，每小节都是1加p-1个0（10…0），末尾再加1。
Φ4(10)=101=101
Φ9(10)=1001001=3×333667
Φ25(10)=100001000010000100001<21>=21401×25601×182521213001<12>
Φ49(10)=1000000100000010000001000000100000010000&#172;001<43>
=505885997×1976730144598190963568023014679333<34>
Φ121(10)= 10000000000100000000001000000000010000000000100000000001000000000010000000000
1000000000010000000000100000000001<111>
=15973×38237×274187
×597149176209530412360795391497657340159943421992502538230831481682232969649167277637825641074323<96>
Φ169(10)= 100000000000010000000000001000000000000100000000000010000000000001000000000000
1000000000000100000000000010000000000001000000000000100000000000010000000000001<157>
=11831×2709246550432452192307438140153898730002165574677<49>
×31198234389876128002952782212916694331889770402831700346290124318006436464972357735278842644511160852523<104>

类型十三：p立方型，(p-1)小节，每小节都是1加p^2-1个0（10…0），末尾再加1。
Φ8(10)=10001=73×137
Φ27(10)=1000000001000000001<19>=3×757×440334654777631<15>
Φ125(10)= 100000000000000000000000010000000000000000000000001000000000000000000000000
10000000000000000000000001<101>
=751×1797655751<10>×176144543406001<15>
×42051775804956304559810859008305819975199677041099230574273451704628125001<74>

类型十四：20,60,100,140,180型，都分成L和M两个φ因子
Φ20L(10)=3541=3541
Φ20M(10)=27961=27961
Φ60L(10)=255522961=61×4188901
Φ60M(10)=39526741=39526741
Φ100L(10)=99004980069800499001<20>=7019801×14103673319201<14>
Φ100M(10)=101005020070200501001<21>=60101×1680588011350901<16>
Φ140L(10)=3572835377...<24>=421×848654483879497562821<21>
Φ140M(10)=2826886473...<25>=3471301×13489841×60368344121<11>
Φ180L(10)=1105097795...<25>=1105097795002994798105101<25>
Φ180M(10)=9048981950...<24>=181×4999437541453012143121<22>

yangchuanju

类型十五：100以内其它类型
Φ16(10)=100000001=17×5882353
Φ18(10)=999001=19×52579
Φ32(10)=10000000000000001<17>=353×449×641×1409×69857
Φ36(10)=999999000001<12>=999999000001<12>
Φ45(10)=999000000999000999999001<24>=238681×4185502830133110721<19>
Φ48(10)=9999999900000001<16>=9999999900000001<16>
Φ50(10)=99999000009999900001<20>=251×5051×78875943472201<14>
Φ54(10)=999999999000000001<18>=70541929×14175966169<11>
Φ63(10)=999000000999000000999999000999999001<36>=10837×23311×45613×45121231×1921436048294281<16>
Φ64(10)=100000000000000000000000000000001<33>=19841×976193×6187457×834427406578561<15>
Φ72(10)=999999999999000000000001<24>=3169×98641×3199044596370769<16>
Φ75(10)=9999900000000009999900000999999999900001<40>=151×4201×15763985553739191709164170940063151<35>
Φ80(10)=99999999000000009999999900000001<32>=5070721×19721061166646717498359681<26>
Φ81(10)=1000000000000000000000000001000000000000000000000000001<55>
=3×163×9397×2462401×676421558270641<15>×130654897808007778425046117<27>
Φ84(10)=1009998990000999899000101<25>=226549×4458192223320340849<19>
Φ88(10)=9999000099990000999900009999000099990001<40>=617×16205834846012967584927082656402106953<38>
Φ96(10)=99999999999999990000000000000001<32>=97×206209×66554101249<11>×75118313082913<14>
Φ98(10)=999999900000009999999000000099999990000001<42>=197×5076141624365532994918781726395939035533<40>
Φ99(10)=999000000999000000999000000999000999999000999999000999999001<60>
=199×397×34849×362853724342990469324766235474268869786311886053883<51>

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唯一周期素数（φ因子）因子渐增
有一类φ因子，它们只有一个，即唯一素数φ因子，又称“唯一周期素数”。
A040017提供了46个唯一素数φ因子（1-45号+46-47号），位号由A051627提供（99个），位数自测。
格式：序号--唯一素数——位号——位数
1-3——1——1位
2-11——2——2位
3-37——3——2位
4-101——4——3位
5-9091——10——4位
6-9901——12——4位
7-333667——9——6位
8-909091——14——6位
9-99990001——24——8位
10-999999000001——36——12位
11-9999999900000001——48——16位
12-909090909090909091——38——18位
13-1111111111111111111——19——19位
14-11111111111111111111111——23——23位
15-900900900900990990990991——39——24位
16-909090909090909090909090909091——62——30位
17-100009999999899989999000000010001——120——33位
18-10000099999999989999899999000000000100001——150——41位
19-9090909090909090909090909090909090909090909090909091——106——52位
20-900900900900900900900900900900990990990990990990990990990991——93——60位
21-909090909090909090909090909090909090909090909090909090909090909091——134——66位
22-142857157142857142856999999985714285714285857142857142855714285571428571428572857143——294——84位
23-999999999999990000000000000099999999999999000000000000009999999999999900000000000001——196——84位

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（接上楼）
24-99999999999999999999999999999999000000000000000000000000000000009999999999999999999999999999999900000000000000000000000000000001——320——128位
25-1098901098901098901098901098901098901098901098901098901098901098901098901098901098901098901098901098901098900989010989010989010989010989010989010989010989010989010989010989010989010989010989010989010989010989010989011——654——217位
26-1000999998999000001000999998999000001000999998999000001000999998999000001000999998999000001000999998999000001000999998998999000001000999998999000001000999998999000001000999998999000001000999998999000001000999998999000001000999998999000001001——738——241位
27-1111099888878889001111100000000000011110998888788890011222109988887889011221109998887788901111110999878890112222210987766677901122222109888790001111110988877889000111221098887888900112221099888878889001111100000000000011110998888788890011111——385——241位
28-9090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909091——586——292位
29-11111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111——317——317位
30-900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990991——597——396位

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（接上楼）
31-1000000000000000000999999999999999999999999999999999998999999999999999999000000000000000000000000000000000001000000000000000000999999999999999999999999999999999998999999999999999999000000000000000000000000000000000000999999999999999999999999999999999998999999999999999999000000000000000000000000000000000001000000000000000000999999999999999999999999999999999998999999999999999999000000000000000000000000000000000001000000000000000001——1404——433位
32-1000000001000000000999999999999999999999999998999999998999999997999999998999999999000000000000000000000000001000000001000000001000000001000000001000000000999999999999999999999999998999999999999999998999999999999999998999999999999999998999999999999999999000000000000000000000000001000000001000000001000000001000000001000000000999999999999999999999999998999999998999999997999999998999999999000000000000000000000000001000000001000000001——945——433位
33-100000000000000000000009999999999999999999999999999999999999999999899999999999999999999990000000000000000000000000000000000000000000100000000000000000000009999999999999999999999999999999999999999999899999999999999999999989999999999999999999999000000000000000000000000000000000000000000010000000000000000000000999999999999999999999999999999999999999999989999999999999999999999000000000000000000000000000000000000000000010000000000000000000001——1452——441位
34-1000000000000000000999999999999999999999999999999999998999999999999999999000000000000000000000000000000000001000000000000000000999999999999999999999999999999999998999999999999999999000000000000000000000000000000000001000000000000000000999999999999999999999999999999999998999999999999999998999999999999999999000000000000000000000000000000000001000000000000000000999999999999999999999999999999999998999999999999999999000000000000000000000000000000000001000000000000000000999999999999999999999999999999999998999999999999999999000000000000000000000000000000000001000000000000000001——1836——577位
35-1000099999998999900000001000099999998999900000001000099999998999900000001000099999998999900000001000099999998999900000001000099999998999900000001000099999998999900000001000099999998999900000001000099999998999900000001000099999998999900000001000099999998999900000001000099999998999900000000999999989999000000010000999999989999000000010000999999989999000000010000999999989999000000010000999999989999000000010000999999989999000000010000999999989999000000010000999999989999000000010000999999989999000000010000999999989999000000010000999999989999000000010000999999989999000000010001——1752——577位

yangchuanju

（接上楼）
36-99009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009901——1172——584位
37-1009998990001009998990001009998990001009998990001009998990001009998990001009998990001009998990001009998990001009998990001009998990001009998990001009998990001009998990001009998990001009998990001009998990001009998990001009998990001009998990001009998990001009998990001009998990001009998990001009998990000999899000100999899000100999899000100999899000100999899000100999899000100999899000100999899000100999899000100999899000100999899000100999899000100999899000100999899000100999899000100999899000100999899000100999899000100999899000100999899000100999899000100999899000100999899000100999899000100999899000101——1812——601位
38-9090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909091——1282——640位
39-1099999999999999999999988999999890000000000000110000001100000010999998899999988999999890000009900000110000001099999900999999009999989000000990000009900000098999990099999900999999010000100100000990000009899998998999989989999901000010010000100100001000999899899998998999989990000999900010010000100099990000999900009998999000099990000999900009998999000099990000999900010010000100099990000999899899998998999989990001001000010010000100100000989999899899998998999990099999901000010010000099000000990000009899999009999990099999900999998900000099000000990000011000000109999990099999889999998899999989000001100000011000000109999999999999889999998900000000000000000000011——1426——661位
40-1000000000000999999999999999999999998999999999999000000000000000000000001000000000000999999999999999999999998999999999999000000000000000000000001000000000000999999999999999999999998999999999999000000000000000000000001000000000000999999999999999999999998999999999999000000000000000000000001000000000000999999999999999999999998999999999999000000000000000000000000999999999999999999999998999999999999000000000000000000000001000000000000999999999999999999999998999999999999000000000000000000000001000000000000999999999999999999999998999999999999000000000000000000000001000000000000999999999999999999999998999999999999000000000000000000000001000000000000999999999999999999999998999999999999000000000000000000000001000000000001——2232——721位

yangchuanju

（接上楼）
41-1000000099999999999999999999999999999999999999999899999990000000000000000000000000000000000000000010000000999999999999999999999999999899999989999998999999900000000000000000000000000010000001000000100000009999999999999999999999999998999999899999989999999000000000000010000001000000100000010000001000000099999999999998999999899999989999998999999899999990000000000000100000010000001000000100000009999999999999899999989999998999999899999989999999000000000000010000001000000100000010000001000000099999999999998999999899999989999999000000000000000000000000000100000010000001000000099999999999999999999999999989999998999999899999990000000000000000000000000001000000099999999999999999999999999999999999999999899999990000000000000000000000000000000000000000010000001——1862——757位
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