Base 13: Cycles Sequences: All Cycle Counts

Blog: The First Pixel: Kaprekar's Constant 6174

Summary
[All Digit Sets]
[Odd Digit Sets]
[Even Digit Sets]
Final Numbers
[Fixed Points]
Cycles Lengths
[All Digit Sets]
[Odd Digit Sets]
[Even Digit Sets]
Iteration Frequencies
[Cycle Start]
[Full Cycle]
Maximum Iteration Sequences
[Cycle Start]
[Full Cycle]
Cycles Sequences
[Maximum Length Cycle Count]
[All Cycle Counts] — [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [Base 13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54] [55] [56] [57] [58] [59] [60] [61] [62] [63] [64] [65] [66] [67] [68] [69] [70] [71] [72] [73] [74] [75] [76] [77] [78] [79] [80] [81] [82] [83] [84] [85] [86] [87] [88] [89] [90] [91] [92] [93] [94] [95] [96] [97] [98] [99] [100] [101] [102] [103] [104] [105] [106] [107] [108] [109] [110] [111] [112] [113] [114] [115] [116] [117] [118] [119] [120] [121] [122] [123] [124] [125] [126] [127] [128] [129] [130] [131] [132] [133] [134] [135] [136] [137] [138] [139] [140] [141] [142] [143] [144] [145] [146] [147] [148] [149] [150] [151] [152] [153] [154] [155] [156] [157] [158] [159] [160] [161] [162] [163] [164] [165] [166] [167] [168] [169] [170] [171] [172] [173] [174] [175] [176] [177] [178] [179] [180] [181] [182] [183] [184] [185] [186] [187] [188] [189] [190] [191] [192] [193] [194] [195] [196] [197] [198] [199] [200] [201] [202] [203] [204] [205] [206] [207] [208] [209] [210] [211] [212] [213] [214] [215] [216] [217] [218] [219] [220] [221] [222] [223] [224] [225] [226] [227] [228] [229] [230] [231] [232] [233] [234] [235] [236] [237] [238] [239] [240] [241] [242] [243] [244] [245] [246] [247] [248] [249] [250] [251] [252] [253] [254] [255]

Radix Character Encoding
0	1	2	3	4	5	6	7	8	9	A	B	C

Color Key
Orange	Fixed Point	number that becomes itself after one iteration
Green	Final Numbers	all numbers eventually reached, from all cycles
Purple	Cycle Start	iterations include entry into cycle, assuming it repeats
Red	Full Cycle	iterations include entire cycle, proving it repeats
Blue	Start Numbers	subset (optimization) of all numbers; minimal for full coverage
Gray	No Results	any calculation that has no results

Digits	# Full Cycles (excluding zero)	Frequency	Cycle Length	Full Cycles (excluding zero) (bold = exactly one cycle [excluding zero])
1	0 cycles	-	-	-
2	1 cycle	72×	(max) 3 nodes	⮎ 1B → 93 → 57 ⮌
3	1 cycle	442×	(max) 2 nodes	⮎ 5C7 → 6C6 ⮌
4	3 cycles	767×	(max) 6 nodes	⮎ 5298 → 7296 → 70B6 → B0B2 → B832 → 9474 ⮌
		704×	(max) 6 nodes	⮎ 30BA → B652 → 90B4 → B472 → 9294 → 7476 ⮌
		336×	3 nodes	⮎ 50B8 → B292 → 9654 ⮌
5	1 cycle	6,175×	(max) 4 nodes	⮎ 83C85 → 92C94 → A4C73 → 95C64 ⮌
6	1 cycle	18,551×	(fixed) (max) 1 node	⮎ 951A74 ⮌
7	3 cycles	27,404×	2 nodes	⮎ 972C954 → A53C873 ⮌
		21,318×	(max) 12 nodes	⮎ 951CA74 → B63C862 → A81CA43 → B75C652 → A61CA63 → B73C852 → A82C943 → A74C753 → 961CA64 → B62C962 → A92C933 → A75C653 ⮌
		1,653×	2 nodes	⮎ A71CA53 → B74C752 ⮌
8	2 cycles	104,016×	(fixed) 1 node	⮎ 9541A874 ⮌
8	2 cycles	21,941×	(max) 4 nodes	⮎ 641CCA87 → B840B842 → B9438832 → 96538764 ⮌
9	3 cycles	256,298×	(max) 5 nodes	⮎ A861CA643 → B761CA652 → B950CB732 → C983C8431 → B963C8632 ⮌
		28,834×	(max) 5 nodes	⮎ A751CA753 → B751CA752 → B951CA732 → B973C8532 → A862C9643 ⮌
		8,785×	(max) 5 nodes	⮎ 9620CBA64 → C962C9631 → BA62C9622 → A982C9433 → A764C7653 ⮌
10	4 cycles	346,542×	(fixed) 1 node	⮎ 95441A8874 ⮌
		185,460×	5 nodes	⮎ 99650B7634 → B651CCA762 → BA640B8622 → B983CC8432 → A984CC7433 ⮌
		97,774×	(max) 9 nodes	⮎ 86661A6665 → 92CCCCCC94 → A832CC9943 → A9750B7533 → B7621AA652 → A881CCA443 → B965CC6632 → A962CC9633 → A973299533 ⮌
		16,857×	4 nodes	⮎ A8630B9643 → B763CC8652 → A9620BA633 → B875CC6542 ⮌
11	1 cycle	1,352,065×	(max) 14 nodes	⮎ A8520CBA743 → C9762C96531 → BA640CB8622 → C9961CA6331 → BB762C96512 → B9940CB8332 → C9864C76431 → B9540CB8732 → C9852C97431 → BA652C97622 → A9840CB8433 → C8753C87541 → B9431CA9832 → B9764C76532 ⮌
12	3 cycles	2,284,802×	(max) 10 nodes	⮎ 96543CC88764 → 985320BA9744 → B86541A87642 → A8441CCA8843 → B9663CC86632 → A9831CCA9433 → B97750B75532 → B96221AAA632 → A8885CC64443 → 986431A98644 ⮌
		416,101×	(fixed) 1 node	⮎ 954441A88874 ⮌
		3,240×	2 nodes	⮎ AA751CCA7523 → BA754CC77522 ⮌
13	3 cycles	4,735,472×	(max) 10 nodes	⮎ A86320CBA9643 → C97751CA75531 → BB7421CAA8512 → BA9862C964322 → A98651CA76433 → B77530CB97552 → C98421CAA8431 → BB8753C875412 → B98431CA98432 → B97653C876532 ⮌
		404,858×	(max) 10 nodes	⮎ A76530CB97653 → C87420CBA8541 → CB9642C986311 → BB9651CA76312 → BA9740CB85322 → C99752C975331 → BA6641CA86622 → B9861CCCA6432 → BA9742C985322 → A98652C976433 ⮌
		59,957×	2 nodes	⮎ A84321CAA9843 → B87753C875542 ⮌
14	3 cycles	8,409,595×	(max) 15 nodes	⮎ A8441CCCCA8843 → B98861CCA64432 → BA86521AA76422 → A88841CCA84443 → B96643CC886632 → A98521CCAA7433 → BA77640B865522 → B98421CCAA8432 → BA9764CC765322 → AA86410BB86423 → BA8743CC885422 → AA86430B986423 → B87643CC886542 → A974320BA98533 → B876641A866542 ⮌
		1,218,334×	(fixed) 1 node	⮎ 9544441A888874 ⮌
		29,758×	(fixed) 1 node	⮎ A97542CC987533 ⮌
15	2 cycles	17,301,794×	(max) 14 nodes	⮎ A966530CB976633 → C87640CCCB86541 → CB88621CAA64411 → BBA8641CA864212 → BA98641CA864322 → B987541CA875432 → B975431CA987532 → B976531CA976532 → B976420CBA86532 → C997530CB975331 → CB86641CA866411 → BBA641CCCA86212 → BBA9743C8853212 → B997642C9865332 ⮌
15	2 cycles	82,053×	2 nodes	⮎ A844321CAA98843 → B877543C8875542 ⮌
16	3 cycles	24,168,112×	(max) 11 nodes	⮎ 9988621CCAA64434 → BA765521AA776522 → A888221CCAAA4443 → BA87663CC8665422 → AA86320CCBA96423 → CA977631A9655321 → B986621CCAA66432 → BA97631CCA965322 → BA977430B9855322 → B9866421AA866432 → A88643CCCC886443 ⮌
		3,669,958×	8 nodes	⮎ 7544443CC8888876 → 9844431CCA988844 → B965543CC8877632 → A9853310BB997433 → BA766641A8666522 → A8841CCCCCCA8443 → B9888421AA844432 → A886444388886443 ⮌
		2,583,672×	(fixed) 1 node	⮎ 95444441A8888874 ⮌
17	4 cycles	48,660,121×	(max) 23 nodes	⮎ A974430CCCB988533 → C9986540CB8764331 → CB865430CB9876411 → CBA85430CB9874211 → CBA97542C98753211 → BB986531CA9764312 → BA976530CB9765322 → C9976420CBA865331 → CB976530CB9765311 → CBA86420CBA864211 → CBA98641CA8643211 → BBA87541CA8754212 → BA986431CA9864322 → B9876541CA8765432 → B9754320CBA987532 → C9976531CA9765331 → BB766420CBA866512 → CA99630CCCB963321 → CBA98732C99543211 → BB986652C97664312 → B996530CCCB976332 → CA988432C99844321 → BA866543C88766422 ⮌
		3,128,347×	17 nodes	⮎ B9965430CB9876332 → C9866530CB9766431 → CB86530CCCB976411 → CBB97631CA9653111 → BBAA7640CB8652212 → CA998630CB9643321 → CB976651CA7665311 → BBA7420CCCBA85212 → CBA99752C97533211 → BB986641CA8664312 → BA97541CCCA875322 → BAA86540CB8764222 → C9986430CB9864331 → CB866541CA8766411 → BBA6430CCCB986212 → CBA98742C98543211 → BB986542C98764312 ⮌
		101,850×	2 nodes	⮎ A8444321CAA988843 → B8775443C88875542 ⮌
		5,604×	2 nodes	⮎ AAA7551CCCA775223 → BAA7554CCC7775222 ⮌
18	2 cycles	81,331,068×	(max) 12 nodes	⮎ 954441CCCCCCA88874 → B8888730CCB9544442 → CA8754430B98875421 → CA8644421AA8886421 → B9884443CC88884432 → A9854443CC88887433 → 997544430B98887534 → B66544421AA8887662 → A884441CCCCA888443 → B988663CCCC8664432 → A99873221AAA954333 → 988766641A86665444 ⮌
18	2 cycles	5,162,144×	(fixed) 1 node	⮎ 954444441A88888874 ⮌
19	2 cycles	140,964,394×	(max) 8 nodes	⮎ BA9754320CBA9875322 → C99876531CA97654331 → BB7665320CBA9766512 → CA997420CCCBA853321 → CBAA87641CA86542211 → BBA886430CB98644212 → CA9975441CA88753321 → BB8765431CA98765412 ⮌
19	2 cycles	156,118×	2 nodes	⮎ A84444321CAA9888843 → B87754443C888875542 ⮌
20	5 cycles	143,711,009×	(max) 12 nodes	⮎ 98544441CCCCA8888744 → B88865430CCB98764442 → CA87544320BA98875421 → CA88644421AA88864421 → B98844443CC888884432 → A98544443CC888887433 → 9975444430B988887534 → B665444421AA88887662 → A8844441CCCCA8888443 → B9886643CCCC88664432 → A998752221AAAA754333 → 9888766421AA86654444 ⮌
		53,770,927×	3 nodes	⮎ BA9765421CCAA8765322 → BA98764310BB98654322 → BA9866441CCA88664322 ⮌
		20,332,911×	7 nodes	⮎ 9999654310BB98763334 → BA6666541CCA87666622 → BA96520CCCCCCBA76322 → CAAA975530B977532221 → CA88866221AAA6644421 → B9888443CCCC88844432 → A9987543CCCC88754333 ⮌
		7,729,548×	(fixed) 1 node	⮎ 9544444441A888888874 ⮌
		248,432×	(fixed) 1 node	⮎ BA9775431CCA98755322 ⮌
21	4 cycles	302,575,243×	(max) 18 nodes	⮎ BBA754320CCCBA9875212 → CBA9986540CB876433211 → CBA9765430CB987653211 → CBA9765320CBA97653211 → CBA9876420CBA86543211 → CBA9875430CB987543211 → CBA9765431CA987653211 → BBA8765320CBA97654212 → CA99875320CBA97543321 → CB98766531CA976654311 → BBA765320CCCBA9765212 → CBA9985530CB977433211 → CBA9766531CA976653211 → BBA876420CCCBA8654212 → CBA9976521CAA76533211 → BBA8876420CBA86544212 → CA99864430CB988643321 → CB97665441CA887665311 ⮌
		51,678,967×	7 nodes	⮎ BA9754320CCCBA9875322 → CAA9876440CB886543221 → CB98754430CB988754311 → CBA8654431CA988764211 → BBA8754430CB988754212 → CA99754431CA988753321 → BB87665431CA987665412 ⮌
		345,578×	3 nodes	⮎ BBB9765330CB997653112 → CAA9866420CBA86643221 → CB9887541CCCA87544311 ⮌
		217,519×	2 nodes	⮎ A844444321CAA98888843 → B877544443C8888875542 ⮌
22	4 cycles	336,170,600×	(max) 12 nodes	⮎ 985444441CCCCA88888744 → B888654430CCB988764442 → CA875444320BA988875421 → CA886444421AA888864421 → B988444443CC8888884432 → A985444443CC8888887433 → 99754444430B9888887534 → B6654444421AA888887662 → A88444441CCCCA88888443 → B98866443CCCC888664432 → A9987542221AAAA8754333 → 98887664421AA886654444 ⮌
		114,610,090×	(fixed) 1 node	⮎ 95444444441A8888888874 ⮌
		97,564,253×	7 nodes	⮎ BA97654421CCAA88765322 → BA987653310BB997654322 → BA98666421CCAA86664322 → BA9876521CCCCAA7654322 → BAAA865531CCA977642222 → BA988764310BB986544322 → BA98664441CCA888664322 ⮌
		9,084×	2 nodes	⮎ AAAA75551CCCCA77752223 → BAAA75554CCCC777752222 ⮌
23	4 cycles	773,075,258×	(fixed) 1 node	⮎ CBA98765320CBA976543211 ⮌
		37,644,362×	(max) 15 nodes	⮎ BA665320CCCCCCCBA976622 → CAAA9765542C98776532221 → BA888653220CBAA97644422 → C9988754430CB9887544331 → CB866544431CA9888766411 → BBA7544430CCCB988875212 → CBA98765330CB9976543211 → CBA97665320CBA976653211 → CBA9876420CCCBA86543211 → CBBA9866421CAA866432111 → BBAA887541CCCA875442212 → BBA99666330CB9966633212 → CA9976662CCCCC966653321 → BAA99743330CB9998533222 → C9987666652C97666654331 ⮌
		23,425,576×	6 nodes	⮎ BBA7544320CCCBA98875212 → CBA99866430CB9866433211 → CBA9766541CCCA876653211 → BBBA8654210CBBA87642112 → CAAA9986430CB9864332221 → CB988766541CA8766544311 ⮌
		306,591×	2 nodes	⮎ A8444444321CAA988888843 → B8775444443C88888875542 ⮌
24	3 cycles	1,117,934,436×	(max) 12 nodes	⮎ 9854444441CCCCA888888744 → B8886544430CCB9888764442 → CA8754444320BA9888875421 → CA8864444421AA8888864421 → B9884444443CC88888884432 → A9854444443CC88888887433 → 997544444430B98888887534 → B66544444421AA8888887662 → A884444441CCCCA888888443 → B988664443CCCC8888664432 → A99875442221AAAA88754333 → 988876644421AA8886654444 ⮌
		113,689,940×	8 nodes	⮎ 98886665421CCAA876664444 → BA66544320CCCCBA98876622 → CAAA87653220BAA976542221 → CA888886421CCAA864444421 → BBA866444421AA8888664212 → AA88844443CCCC8888844423 → A998664443CCCC8888664333 → 999965542221AAAA87763334 ⮌
		20,053,311×	(fixed) 1 node	⮎ 954444444441A88888888874 ⮌
25	5 cycles	1,290,137,646×	(max) 8 nodes	⮎ BBA76544320CCCBA988765212 → CBA998664320CBA9866433211 → CBA98766541CCCA8766543211 → BBBA86643210CBBA986642112 → CAAA9987541CCCA8754332221 → BBAA87765430CB98765542212 → CA9987543220CBAA987543321 → CB9887665431CA98766544311 ⮌
		322,856,847×	7 nodes	⮎ BA9875444320CBA9888754322 → C99876544431CA98887654331 → BB7665444320CBA9888766512 → CA997544320CCCBA988753321 → CBAA87755430CB98775542211 → CBA987543221CAAA987543211 → BBA888765431CA98765444212 ⮌
		232,444,894×	(fixed) 1 node	⮎ CBA987654320CBA9876543211 ⮌
		6,668,693×	(fixed) 1 node	⮎ CCBA98765320CBA9765432101 ⮌
		374,903×	2 nodes	⮎ A84444444321CAA9888888843 → B87754444443C888888875542 ⮌
26	3 cycles	1,648,065,396×	(max) 12 nodes	⮎ 98544444441CCCCA8888888744 → B88865444430CCB98888764442 → CA87544444320BA98888875421 → CA88644444421AA88888864421 → B98844444443CC888888884432 → A98544444443CC888888887433 → 9975444444430B988888887534 → B665444444421AA88888887662 → A8844444441CCCCA8888888443 → B9886644443CCCC88888664432 → A998754442221AAAA888754333 → 9888766444421AA88886654444 ⮌
		1,026,230,307×	2 nodes	⮎ BA988876521CCCCAA765444322 → BAAA86664321CCAA9866642222 ⮌
		33,179,432×	(fixed) 1 node	⮎ 9544444444441A888888888874 ⮌
27	6 cycles	2,124,518,460×	(max) 5 nodes	⮎ BBAAA8876530CCCB97654422212 → CBA9997664320CBA98665333211 → CBA987666530CCCB97666543211 → CBBA88644210CCCBBA886442111 → CBBBA99766421CAA86653321111 ⮌
		1,739,172,760×	(fixed) 1 node	⮎ CBA9876544320CBA98876543211 ⮌
		39,904,610×	(fixed) 1 node	⮎ CCBA987654320CBA98765432101 ⮌
		6,668,693×	(fixed) 1 node	⮎ CCCBA98765320CBA97654321001 ⮌
		519,472×	2 nodes	⮎ A844444444321CAA98888888843 → B877544444443C8888888875542 ⮌
		13,428×	2 nodes	⮎ AAAAA755551CCCCCA7777522223 → BAAAA755554CCCCC77777522222 ⮌
28	4 cycles	5,283,588,500×	(max) 12 nodes	⮎ 985444444441CCCCA88888888744 → B888654444430CCB988888764442 → CA875444444320BA988888875421 → CA886444444421AA888888864421 → B988444444443CC8888888884432 → A985444444443CC8888888887433 → 99754444444430B9888888887534 → B6654444444421AA888888887662 → A88444444441CCCCA88888888443 → B98866444443CCCC888888664432 → A9987544442221AAAA8888754333 → 98887664444421AA888886654444 ⮌
		253,380,191×	7 nodes	⮎ 99996554443310BB998887763334 → BA66666544421CCAA88876666622 → BA98664320CCCCCCCCBA98664322 → CAAA9988554410BB887744332221 → CAA88866444421AA888866444221 → B98884444443CCCC888888844432 → A99875444443CCCC888888754333 ⮌
		49,823,192×	(fixed) 1 node	⮎ 95444444444441A8888888888874 ⮌
		61,584×	(fixed) 1 node	⮎ AA9977554432CCCC998877553323 ⮌
29	9 cycles	3,908,442,971×	(max) 5 nodes	⮎ BBAAA88766530CCCB976654422212 → CBA99976643210CBBA98665333211 → CBAA987666530CCCB976665432211 → CBBA987644210CCCBBA8865432111 → CBBBA998665321CAA976643321111 ⮌
		2,625,975,232×	3 nodes	⮎ CBA9888765320CCCBA97654443211 → CBBA9876553320CBA997765432111 → CBAA9876653220CBAA97665432211 ⮌
		1,175,460,587×	(fixed) 1 node	⮎ CBA98765444320CBA988876543211 ⮌
		129,789,476×	(fixed) 1 node	⮎ CCBA9876544320CBA988765432101 ⮌
		39,904,610×	(fixed) 1 node	⮎ CCCBA987654320CBA987654321001 ⮌
		10,801,338×	2 nodes	⮎ CBAA988766420CCCBA86654432211 → CBBA9976653220CBAA97665332111 ⮌
		6,668,693×	(fixed) 1 node	⮎ CCCCBA98765320CBA976543210001 ⮌
		984,160×	(fixed) 1 node	⮎ BBA9877554331CCCA998775543212 ⮌
		627,840×	2 nodes	⮎ A8444444444321CAA988888888843 → B8775444444443C88888888875542 ⮌
30+	-	-	-	Not showing digit sets 30+ due to verbosity.