Base 106: Iteration Frequencies: Full Cycle

Blog: The First Pixel: Kaprekar's Constant 6174

Radix Character Encoding
0123456789ABCDEF
GHIJKLMNOPQRSTUV
WXYZabcdefghijkl
mnopqrstuvwxyzΑΒ
ΓΔΕΖΗΘΙΚΛΜΝΞΟΠΡΣ
ΤΥΦΧΨΩαβγδεζηθικ
λμνξοπρςστυφχψωА

Color Key
OrangeFixed Pointnumber that becomes itself after one iteration
GreenFinal Numbersall numbers eventually reached, from all cycles
PurpleCycle Startiterations include entry into cycle, assuming it repeats
RedFull Cycleiterations include entire cycle, proving it repeats
BlueStart Numberssubset (optimization) of all numbers; minimal for full coverage
GrayNo Resultsany calculation that has no results

Digits 1 2 3 4 5 6
Average Steps
Full Cycle		1.99 step
avg		53.51 step
avg		22.53 step
avg		28.33 step
avg		24.82 step
avg		277.72 step
avg
Step Frequency
Full Cycle
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1×
105×
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105×
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0×
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27×
2,782×
2,756×
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0×
1×
2,967×
2,860×
5,718×
5,710×
5,698×
5,682×
5,662×
5,638×
5,610×
5,578×
5,542×
5,502×
5,458×
5,410×
5,358×
5,302×
5,242×
5,178×
5,110×
5,038×
4,962×
4,882×
4,798×
4,710×
4,618×
4,522×
4,422×
4,318×
4,210×
4,098×
3,982×
3,862×
3,738×
3,610×
3,478×
3,342×
3,202×
3,058×
2,910×
2,758×
2,602×
2,442×
2,278×
2,110×
1,938×
1,762×
1,582×
1,398×
1,210×
1,018×
822×
622×
418×
210×
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0×
1×
105×
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13,044×
38,410×
151,580×
10,281×
6,888×
0×
0×
0×
0×
0×
0×
0×
0×
326,256×
908,640×
3,637,920×
236,944×
210,077×
13,113×
9,992×
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0×
1×
105×
8,478×
142,404×
321,882×
263,982×
398,558×
715,860×
937,478×
1,372,098×
1,471,926×
2,094,434×
1,785,676×
2,860,462×
2,154,250×
3,824,106×
2,519,360×
4,838,336×
3,222,752×
5,853,630×
4,157,134×
6,865,126×
5,160,404×
7,099,770×
5,467,398×
7,113,288×
5,122,756×
6,624,452×
4,618,860×
6,094,338×
4,154,424×
5,132,084×
3,741,384×
4,005,956×
2,950,522×
2,907,096×
2,011,858×
1,689,770×
1,134,300×
763,438×
453,568×
218,950×
86,590×
29,878×
2,400×
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0×
1×
2,967×
0×
0×
0×
82,816×
138,274×
208,360×
300,356×
354,986×
117,462×
188,806×
239,764×
158,072×
224,282×
109,614×
66,110×
120,508×
223,590×
225,366×
355,604×
788,734×
646,087×
567,213×
633,230×
874,965×
896,213×
1,567,253×
1,987,766×
1,792,063×
1,852,455×
2,474,605×
2,845,123×
2,425,095×
2,386,200×
2,437,520×
2,280,012×
2,255,927×
1,826,246×
1,686,194×
1,499,029×
1,627,107×
1,353,545×
1,261,728×
1,526,714×
943,324×
969,580×
903,071×
706,461×
1,253,688×
1,131,971×
1,095,295×
1,487,347×
1,061,535×
901,466×
815,486×
982,731×
464,692×
620,432×
490,522×
668,234×
571,414×
799,250×
400,312×
152,292×
194,716×
220,446×
363,478×
461,604×
472,412×
458,230×
638,960×
565,496×
578,944×
746,110×
839,463×
668,397×
519,125×
456,290×
464,937×
543,616×
560,989×
483,984×
552,515×
848,276×
832,879×
773,910×
790,284×
312,937×
328,702×
442,664×
435,056×
575,526×
547,932×
817,903×
1,028,333×
807,919×
953,660×
1,295,265×
1,667,054×
1,729,386×
1,641,772×
1,464,627×
1,446,124×
1,666,894×
2,156,979×
1,939,868×
2,952,479×
4,870,117×
3,850,665×
3,199,343×
3,539,418×
4,439,477×
4,603,899×
4,318,309×
4,094,304×
3,292,396×
3,026,328×
2,625,173×
3,483,208×
3,790,871×
3,755,014×
3,365,153×
3,161,572×
2,847,109×
2,724,027×
2,973,936×
3,142,286×
4,147,240×
4,037,517×
3,944,376×
3,762,556×
3,772,371×
3,960,623×
3,222,516×
3,099,858×
2,724,586×
3,119,264×
2,988,117×
3,115,121×
2,472,446×
2,466,402×
3,628,621×
4,702,292×
4,936,189×
4,649,536×
3,789,329×
2,924,240×
3,422,598×
3,926,653×
3,327,561×
3,009,246×
4,437,486×
7,579,007×
6,404,304×
6,844,232×
7,776,362×
7,713,156×
7,622,763×
7,871,061×
7,810,720×
6,973,429×
5,892,071×
5,828,744×
6,984,278×
7,311,922×
5,886,871×
5,129,556×
4,535,973×
4,562,970×
4,969,695×
5,138,649×
5,194,574×
5,558,552×
4,717,581×
4,755,134×
5,655,438×
5,138,926×
5,038,467×
5,113,505×
5,646,253×
7,236,229×
6,409,811×
5,745,651×
4,815,523×
4,900,140×
5,032,313×
5,032,958×
5,742,476×
5,514,868×
6,561,874×
5,849,874×
6,033,456×
5,524,904×
5,155,096×
4,937,389×
4,300,225×
4,286,593×
4,717,984×
4,310,720×
4,262,599×
3,654,274×
4,242,183×
3,984,427×
4,000,036×
4,078,034×
3,632,524×
3,361,525×
3,262,637×
3,062,286×
2,951,077×
3,245,460×
3,510,468×
3,262,377×
3,010,038×
2,995,345×
3,195,765×
2,784,109×
2,833,068×
2,747,396×
2,605,623×
2,866,434×
3,274,935×
3,160,636×
2,884,051×
2,616,624×
3,631,158×
2,987,890×
3,171,073×
3,714,402×
3,169,339×
3,608,071×
3,124,003×
2,803,090×
3,029,130×
3,598,991×
3,593,436×
3,328,501×
2,968,953×
2,564,839×
2,341,535×
2,071,540×
2,235,720×
2,567,375×
2,972,814×
2,425,419×
1,999,213×
2,237,152×
2,030,200×
2,601,859×
2,781,301×
2,421,859×
2,151,145×
5,114,739×
9,706,567×
9,164,086×
11,286,276×
12,223,905×
11,690,150×
12,043,872×
11,553,692×
11,583,017×
12,195,561×
13,474,376×
12,768,332×
13,823,410×
14,350,640×
13,581,785×
13,958,421×
13,931,352×
13,837,822×
14,248,875×
13,134,732×
14,595,275×
14,255,979×
13,645,075×
15,508,356×
16,448,100×
15,494,708×
15,723,151×
16,218,009×
15,647,519×
14,488,278×
14,286,244×
13,176,042×
13,567,787×
13,077,343×
13,633,180×
13,584,537×
13,118,367×
12,529,182×
13,701,988×
12,341,958×
11,289,179×
11,023,050×
11,013,906×
10,994,813×
9,731,568×
9,414,477×
10,296,128×
10,701,628×
11,402,424×
11,104,941×
10,859,606×
12,348,363×
12,046,618×
12,588,954×
12,235,610×
12,143,139×
12,609,224×
11,479,042×
10,915,720×
10,539,234×
10,513,734×
10,913,897×
10,569,724×
10,869,647×
11,640,610×
11,393,822×
9,150,450×
8,315,415×
8,241,756×
8,383,808×
8,316,829×
8,036,351×
8,377,535×
8,545,145×
8,149,475×
7,463,216×
6,884,001×
8,169,186×
8,675,810×
9,897,537×
10,338,458×
10,422,134×
10,132,860×
10,182,698×
9,904,643×
10,198,454×
10,178,067×
10,314,405×
9,140,969×
9,733,028×
9,811,904×
8,191,028×
8,209,974×
9,840,232×
10,231,585×
9,435,003×
8,144,081×
8,324,849×
8,378,236×
7,890,741×
7,490,752×
8,131,803×
7,019,862×
6,304,655×
5,939,398×
6,120,955×
5,987,082×
6,650,511×
6,666,934×
6,878,661×
6,487,115×
6,911,691×
6,088,939×
6,149,331×
6,436,362×
5,136,385×
5,711,316×
6,832,715×
6,670,539×
7,264,047×
6,950,539×
5,756,964×
5,711,644×
5,804,454×
6,358,961×
6,278,882×
7,047,381×
6,009,500×
5,213,589×
4,834,086×
4,451,966×
3,599,029×
3,954,735×
4,997,490×
4,806,280×
4,625,962×
4,518,453×
4,673,070×
4,005,222×
4,671,064×
4,209,291×
4,690,779×
4,240,247×
3,534,936×
3,348,459×
3,655,176×
3,470,809×
3,354,535×
3,189,854×
2,554,923×
3,730,405×
3,646,604×
3,695,179×
2,974,315×
2,683,604×
2,733,115×
3,112,447×
3,356,847×
3,179,693×
3,228,911×
3,456,729×
3,317,555×
3,292,461×
3,636,790×
3,491,967×
3,307,091×
3,232,734×
2,936,949×
2,512,175×
2,195,827×
2,132,503×
1,864,436×
1,698,444×
1,856,440×
2,224,763×
1,936,906×
2,039,725×
1,983,394×
2,090,043×
1,863,256×
1,552,503×
1,752,476×
1,330,124×
1,418,030×
1,914,282×
1,461,593×
1,569,458×
2,373,435×
1,704,206×
1,959,746×
2,452,677×
2,440,884×
2,482,216×
2,451,866×
2,355,204×
1,714,804×
1,498,455×
1,279,406×
1,108,093×
835,324×
718,117×
745,724×
830,014×
893,356×
803,740×
877,486×
395,274×
501,824×
483,029×
529,136×
788,587×
1,073,625×
649,660×
331,540×
619,554×
900,960×
1,193,400×
1,315,298×
1,348,184×
1,210,180×
1,261,166×
1,085,497×
1,302,137×
1,463,156×
1,491,049×
1,295,958×
1,804,075×
1,890,010×
1,447,229×
1,196,522×
1,112,119×
1,131,544×
1,075,820×
963,030×
816,207×
715,622×
718,608×
785,084×
1,392,953×
1,322,287×
1,187,362×
887,814×
867,842×
625,856×
736,255×
483,411×
445,820×
225,092×
293,920×
582,951×
660,472×
899,573×
787,964×
396,039×
641,952×
664,059×
355,802×
501,828×
676,666×
799,660×
788,689×
628,818×
592,432×
456,604×
387,789×
315,962×
318,222×
101,466×
178,498×
219,700×
75,516×
69,172×
92,784×
79,566×
66,844×
109,572×
186,528×
173,247×
210,396×
135,648×
92,802×