A151959 -id:A151959 - cp's OEIS frontend

A151949 a(n) = image of n under the Kaprekar map n -> (n with digits sorted into descending order) - (n with digits sorted into ascending order).

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 9, 0, 9, 18, 27, 36, 45, 54, 63, 72, 18, 9, 0, 9, 18, 27, 36, 45, 54, 63, 27, 18, 9, 0, 9, 18, 27, 36, 45, 54, 36, 27, 18, 9, 0, 9, 18, 27, 36, 45, 45, 36, 27, 18, 9, 0, 9, 18, 27, 36, 54, 45, 36, 27, 18, 9, 0, 9, 18, 27, 63, 54, 45, 36, 27, 18, 9, 0, 9, 18, 72, 63, 54, 45, 36, 27, 18, 9, 0, 9, 81, 72, 63, 54, 45, 36, 27, 18, 9, 0, 99, 99, 198, 297, 396, 495, 594, 693, 792, 891, 99, 0, 99, 198, 297, 396, 495, 594, 693, 792

Offset: 0

N. J. A. Sloane, Aug 18 2009

Entries are multiples of 9 - see A151950.

a(n) = A004186(n) - A004185(n); a(A010785(n)) = 0. - Reinhard Zumkeller, corrected: Mar 23 2015, Jul 09 2013

			For n = 15, a(15) = 51 - 15 = 36. - _Indranil Ghosh_, Feb 01 2017

Indranil Ghosh, Table of n, a(n) for n = 0..20000 (terms 0..1000 from Joseph Myers and Robert G. Wilson v)
H. Hanslik, E. Hetmaniok, I. Sobstyl, et al., Orbits of the Kaprekar's transformations-some introductory facts, Zeszyty Naukowe Politechniki Śląskiej, Seria: Matematyka Stosowana z. 5, Nr kol. 1945; 2015.
M. Kauers and C. Koutschan, Some D-finite and some possibly D-finite sequences in the OEIS, arXiv:2303.02793 [cs.SC], 2023.
R. J. Mathar, Maple code for A151949 and A151959
Joseph Myers, C program for computing sequences related to the Kaprekar map
Joseph Myers, List of cycles under Kaprekar map (all numbers with <= 60 digits; cycles are represented by their smallest value)
Index entries for the Kaprekar map

In other bases: A164884 (base 2), A164993 (base 3), A165012 (base 4), A165032 (base 5), A165051 (base 6), A165071 (base 7), A165090 (base 8), A165110 (base 9). - Joseph Myers, Sep 05 2009
Cf. A151959, A069746, A347688.
Cf. also A004185, A004186, A099009 (fixed points).

a151949 n = a004186 n - a004185 n
-- Reinhard Zumkeller, corrected: Mar 23 2015, Jul 09 2013

f[n_] := Module[{idn = IntegerDigits@n, idns}, idns = Sort@ idn; FromDigits@ Reverse@ idns - FromDigits@ idns]; Table[ f@n, {n, 0, 200}] (* Harvey P. Dale, Aug 18 2009 *)
Flatten[Table[Differences[FromDigits /@ {y = Sort[x = IntegerDigits[n]], Reverse[y]}], {n, 0, 74}]] (* Jayanta Basu, Jul 11 2013 *)

a(n) = {my(d=digits(n)); fromdigits(vecsort(d,,4)) - fromdigits(vecsort(d));} \\ Michel Marcus, Dec 08 2019

def A151949(n):
    return int("".join(sorted(str(n),reverse=True)))-int("".join(sorted(str(n)))) # Indranil Ghosh, Feb 01 2017

More terms from Robert G. Wilson v, Aug 19 2009

More than the usual number of terms are shown in order to distinguish this from similar sequences. - N. J. A. Sloane, Sep 22 2021

A099009 Fixed points of the Kaprekar mapping f(n) = n' - n'', where in n' the digits of n are arranged in descending, in n'' in ascending order.

Original entry on oeis.org

0, 495, 6174, 549945, 631764, 63317664, 97508421, 554999445, 864197532, 6333176664, 9753086421, 9975084201, 86431976532, 555499994445, 633331766664, 975330866421, 997530864201, 999750842001, 8643319766532, 63333317666664

Offset: 1

Klaus Brockhaus, Sep 22 2004

There are no seven-digit fixed points.
Let d(n) denote n repetitions of the digit d. The sequence includes the following for all n>=0: 5(n)499(n)4(n)5, 63(n)176(n)4, 8643(n)1976(n)532. - Jens Kruse Andersen, Oct 04 2004
0's in n giving leading 0's in n'' is allowed.
For every natural number n let n' and n" be the numbers obtained by arranging the digits of n into decreasing and increasing order, and let f(n)=n'-n". It is known that the number 6174 is invariant under this transformation and that applying f a certain number of times to a number n with four digits the numbers 0, 495 or 6174 are always reached. - Vincenzo Librandi, Nov 17 2010
Each term of A055162(n) corresponds to A099009(n+1), with its digits being reordered in the ascending manner. - Alexander R. Povolotsky, Apr 27 2012
All terms of this sequence are divisible by nine, a(n)/9 = A132155(n). - Alexander R. Povolotsky, Apr 29 2012
A055160 differs from this sequence only at the positions of two terms in it: 554999445 and 555499994445. - Alexander R. Povolotsky, May 01 2012
The union of the sequences A214555, A214556, A214557, A214558, A214559 and the element 0 gives the sequence A099009. - Syed Iddi Hasan, Jul 24 2012
The comment made by Jens Kruse Andersen is missing one more family of terms (which starts with one or more digits "9" and ends with the digit "1"): 97508421, 9753086421, 9975084201, 975330866421, 997530864201, 999750842001, ... This family could be generalized (using the same method as in Andersen's comment) and it is actually covered by Syed Iddi Hasan in A214559. Also A214557 and A214558 (both - by Syed Iddi Hasan) are variants of Andersen's 8643(n)1976(n)532. - Alexander R. Povolotsky, Mar 14 2015
Fixed points of A151949. - Reinhard Zumkeller, Mar 23 2015

			6174 is a fixed point of the mapping and hence a term: 6174 -> 7641 - 1467 = 6174.

Syed Iddi Hasan, Table of n, a(n) for n = 1..8924
Mauro Fiorentini, Kaprekar (costante di) (in Italian)
Manuj Mishra, Illustration of first 8923 terms, with each digit in a different color
Manuj Mishra, Illustration as above but only including terms of even length
Manuj Mishra, Illustration as above but only including terms of odd length
Joseph Myers, List of cycles under Kaprekar map (all numbers with <= 60 digits; cycles are represented by their smallest value)
Conrad Roche, Kaprekar Series Generator.
Eric Weisstein's World of Mathematics, Kaprekar Routine
Index entries for the Kaprekar map

Cf. A090429, A069746, A099010, A151959, A055162, A132155, A055160, A214557, A214558, A214559
In other bases: A163205 (base 2), A164997 (base 3), A165016 (base 4), A165036 (base 5), A165055 (base 6), A165075 (base 7), A165094 (base 8), A165114 (base 9).
Cf. A151949, A004185, A004186.

a099009 n = a099009_list !! (n-1)
a099009_list = [x | x <- [0..], a151949 x == x]
-- Reinhard Zumkeller, Mar 23 2015

a:=func; [k:k in [0..10^7]|a(k)]; // Marius A. Burtea, Sep 12 2019

f[n_] := Block[{d = IntegerDigits@ n, a, b}, a = FromDigits@ Sort@ d; b = FromDigits@ Reverse@ Sort@ d; n == b - a]; Select[Range@ 1000000, f] (* Michael De Vlieger, Mar 20 2015 *)

# (version 2.4) from Tim Peters
def extend(base, start, n):
    if n == 0:
        yield base
        return
    for i in range(start, 10):
        for x in extend(base + str(i), i, n-1):
            yield x
def drive(n):
    result = []
    for lo in extend("", 0, n):
        ilo = int(lo)
        if ilo == 0 and n > 1:
            continue
        hi = lo[::-1]
        diff = str(int(hi) - ilo)
        diff = "0" * (n - len(diff)) + diff
        if sorted(diff) == list(lo):
            result.append(diff)
    return sorted(result)
for n in range(1, 17):
    # print("Length", n)
    # print('-' * 40)
    for r in drive(n):
        print(r, end=', ')

More terms from Jens Kruse Andersen and Tim Peters (tim(AT)python.org), Oct 04 2004

Corrected by Jens Kruse Andersen, Oct 25 2004

A165047 Consider the base-5 Kaprekar map x->K(x) described in A165032. Sequence gives the smallest number that belongs to a cycle of length n under repeated iteration of this map, or -1 if there is no cycle of length n.

Original entry on oeis.org

0, 48, 45781056, 1992, 7488, 249992, 26648194761946797370910644531056, 170312312, 447082519531056, 953674316406249992, 43487548828124832, 68219378590583801269531056

Offset: 1

Joseph Myers, Sep 04 2009

Known values (to 100 base-5 digits):
a(1) = 0 (base 10) = 0 (base 5)
a(2) = 48 (base 10) = 143 (base 5)
a(3) = 45781056 (base 10) = 43204443211 (base 5)
a(4) = 1992 (base 10) = 30432 (base 5)
a(5) = 7488 (base 10) = 214423 (base 5)
a(6) = 249992 (base 10) = 30444432 (base 5)
a(7) = 26648194761946797370910644531056 (base 10) = 432044444444444444444444444444444444444443211 (base 5)
a(8) = 170312312 (base 10) = 322044443222 (base 5)
a(9) = 447082519531056 (base 10) = 432044444444444443211 (base 5)
a(10) = 953674316406249992 (base 10) = 30444444444444444444444432 (base 5)
a(11) = 43487548828124832 (base 10) = 331044444444444444443312 (base 5)
a(12) = 68219378590583801269531056 (base 10) = 4320444444444444444444444444444443211 (base 5)
a(13) = 388774887899923005107893914100714027881532907485961914056 (base 10) = 432222222222222222222222044444444444444444444444444444443222222222222222222222211 (base 5)
a(14) = 4366040229797363281056 (base 10) = 4320444444444444444444444443211 (base 5)
a(15) = 15550995515996920287582474884401599410921335220336914056 (base 10) = 4322222222222222222222222222222222044444444432222222222222222222222222222222211 (base 5)
a(18) = 1705484464764595031738281056 (base 10) = 432044444444444444444444444444444443211 (base 5)
a(20) = 6505906924303417326882481575012207031056 (base 10) = 432044444444444444444444444444444444444444444444444443211 (base 5)
a(21) = 416378043155418708920478820800781056 (base 10) = 432044444444444444444444444444444444444444444443211 (base 5)
a(22) = 39708904567281599895522958831861615180969238281056 (base 10) = 43204444444444444444444444444444444444444444444444444444444444444443211 (base 5)
a(23) = 16655121726216748356819152832031056 (base 10) = 4320444444444444444444444444444444444444444443211 (base 5)
a(24) = 1479271969470441337508073405618280737883196707116439938545227050781056 (base 10) = 432044444444444444444444444444444444444444444444444444444444444444444444444444444444444444444443211 (base 5)
a(26) = 260236276972136693075299263000488281056 (base 10) = 4320444444444444444444444444444444444444444444444443211 (base 5)
a(27) = 9694556779121484349492909871059964643791317939758300781056 (base 10) = 43204444444444444444444444444444444444444444444444444444444444444444444444444443211 (base 5)
a(28) = 151477449673773192960826716735311947559239342808723449707031056 (base 10) = 43204444444444444444444444444444444444444444444444444444444444444444444444444444444443211 (base 5)
a(29) = 4066191827689635829301550984382629394531056 (base 10) = 4320444444444444444444444444444444444444444444444444444443211 (base 5)
a(30) = 101654795692240895732538774609565734863281056 (base 10) = 432044444444444444444444444444444444444444444444444444444443211 (base 5)
a(33) = 1588356182691263995820918353274464607238769531056 (base 10) = 432044444444444444444444444444444444444444444444444444444444444443211 (base 5)
a(35) = 992722614182039997388073970796540379524230957031056 (base 10) = 4320444444444444444444444444444444444444444444444444444444444444444443211 (base 5)
a(36) = 59170878778817653500322936224731229515327868284657597541809082031056 (base 10) = 4320444444444444444444444444444444444444444444444444444444444444444444444444444444444444444443211 (base 5)
a(39) = 387782271164859373979716394842398585751652717590332031056 (base 10) = 432044444444444444444444444444444444444444444444444444444444444444444444444443211 (base 5)
a(41) = 242363919478037108737322746776499116094782948493957519531056 (base 10) = 4320444444444444444444444444444444444444444444444444444444444444444444444444444443211 (base 5)

Joseph Myers, Table of n, a(n) for n=1..15
Index entries for the Kaprekar map

Cf. A165032, A165036, A165037, A165039, A165041, A165043.

In other bases: A153881 (base 2), A165008 (base 3), A165028 (base 4), A165067 (base 6), A165086 (base 7), A165106 (base 8), A165126 (base 9), A151959 (base 10).

A165126 Consider the base-9 Kaprekar map x->K(x) described in A165110. Sequence gives the smallest number that belongs to a cycle of length n under repeated iteration of this map, or -1 if there is no cycle of length n.

Original entry on oeis.org

0, 16, 2256, 31596672, 34960, 26531651360, 14560721001508880, 8724454714749973651840, 108401672318914272, 711223428647787942432, 16513410921312, 278474880, 4754966263206652084045296, 183696

Offset: 1

Joseph Myers, Sep 04 2009

Known values (to 70 base-9 digits):
a(1) = 0 (base 10) = 0 (base 9)
a(2) = 16 (base 10) = 17 (base 9)
a(3) = 2256 (base 10) = 3076 (base 9)
a(4) = 31596672 (base 10) = 65407433 (base 9)
a(5) = 34960 (base 10) = 52854 (base 9)
a(6) = 26531651360 (base 10) = 75430875432 (base 9)
a(7) = 14560721001508880 (base 10) = 77643208887654212 (base 9)
a(8) = 8724454714749973651840 (base 10) = 87654320888888876543211 (base 9)
a(9) = 108401672318914272 (base 10) = 644444418864444443 (base 9)
a(10) = 711223428647787942432 (base 10) = 6444444441886444444443 (base 9)
a(11) = 16513410921312 (base 10) = 64418888886443 (base 9)
a(12) = 278474880 (base 10) = 641888643 (base 9)
a(13) = 4754966263206652084045296 (base 10) = 65544444218888886644444333 (base 9)
a(14) = 183696 (base 10) = 308876 (base 9)
a(15) = 8780535458788649952 (base 10) = 64444441888864444443 (base 9)
a(16) = 8811048483031324779456676539593726674416 (base 10) = 655444442188888888888888888888886644444333 (base 9)
a(18) = 177097392902234856396140027020301600 (base 10) = 7766644432208888888888888766544422212 (base 9)
a(20) = 50771339309018227821951440 (base 10) = 776444432088888887654444212 (base 9)
a(21) = 124998824875093374012011515622478472976 (base 10) = 7544444444444421888888886644444444444432 (base 9)
a(22) = 31197333902107825741164471552 (base 10) = 655444444444308875444444444333 (base 9)
a(23) = 685322163857921701893212141347733334983278765142051291692546183840 (base 10) = 876544444444444432088888888888888888888888888888887654444444444443211 (base 9)
a(25) = 702826002884083319045760971727413317645857409225018391041723392 (base 10) = 655444444444444444443088888888888888888888887544444444444444444333 (base 9)
a(27) = 94116815581356594318021072974737787790560 (base 10) = 7766644432208888888888888888888766544422212 (base 9)
a(28) = 2156904722606587695378609845389399883833898563216184240 (base 10) = 777777655554444333332222210888776666655555444433332111112 (base 9)
a(29) = 26981972242036651232570366433200 (base 10) = 776444444443208888876544444444212 (base 9)
a(31) = 377773874412068206712875872 (base 10) = 6441888888888888888888886443 (base 9)
a(35) = 2520442200659768347220271484032 (base 10) = 65408888888888888888888888887433 (base 9)
a(39) = 1161485426793562822354375723686123973520 (base 10) = 77644444320888888888888888888876544444212 (base 9)
a(40) = 16327050854444484146838503988925985281714578428892247536 (base 10) = 6554444444444444444444442188888866444444444444444444444333 (base 9)
a(46) = 10150412679066664692018845810715370585139195588212689167959856 (base 10) = 7665444444444444444444444422218888666644444444444444444444443222 (base 9)
a(65) = 26152889553714885216926446303415314330076524215805520 (base 10) = 7654308888888888888888888888888888888888888888888754322 (base 9)

Joseph Myers, Table of n, a(n) for n=1..16
Index entries for the Kaprekar map

Cf. A165110, A165114, A165115, A165117, A165119, A165121.

In other bases: A153881 (base 2), A165008 (base 3), A165028 (base 4), A165047 (base 5), A165067 (base 6), A165086 (base 7), A165106 (base 8), A151959 (base 10).

A165008 Consider the base-3 Kaprekar map x->K(x) described in A164993. Sequence gives the smallest number that belongs to a cycle of length n under repeated iteration of this map, or -1 if there is no cycle of length n.

Original entry on oeis.org

0, 32, 320, 26240, 1240024, 11160256, 2297798771761759543384, 15075857741528904364175224, 8135830264, 5931020266096, 659002251784, 350220815692997944

Offset: 1

Joseph Myers, Sep 04 2009

Known values (to 200 base-3 digits):
a(1) = 0 (base 10) = 0 (base 3)
a(2) = 32 (base 10) = 1012 (base 3)
a(3) = 320 (base 10) = 102212 (base 3)
a(4) = 26240 (base 10) = 1022222212 (base 3)
a(5) = 1240024 (base 10) = 2022222222211 (base 3)
a(6) = 11160256 (base 10) = 202222222222211 (base 3)
a(7) = 2297798771761759543384 (base 10) = 202222222222222222222222222222222222222222211 (base 3)
a(8) = 15075857741528904364175224 (base 10) = 20222222222222222222222222222222222222222222222222211 (base 3)
a(9) = 8135830264 (base 10) = 202222222222222222211 (base 3)
a(10) = 5931020266096 (base 10) = 202222222222222222222222211 (base 3)
a(11) = 659002251784 (base 10) = 2022222222222222222222211 (base 3)
a(12) = 350220815692997944 (base 10) = 2022222222222222222222222222222222211 (base 3)
a(14) = 480412641554176 (base 10) = 2022222222222222222222222222211 (base 3)
a(15) = 1202547548374693105751742636119782969638250884664 (base 10) = 20222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222211 (base 3)
a(18) = 3151987341236981536 (base 10) = 202222222222222222222222222222222222211 (base 3)
a(20) = 1221144477063841253498193544 (base 10) = 202222222222222222222222222222222222222222222222222222211 (base 3)
a(21) = 1675095304614322707130576 (base 10) = 202222222222222222222222222222222222222222222222211 (base 3)
a(22) = 5840696178317563736403001300866976 (base 10) = 20222222222222222222222222222222222222222222222222222222222222222222211 (base 3)
a(23) = 186121700512702523014504 (base 10) = 2022222222222222222222222222222222222222222222211 (base 3)
a(24) = 133616394263854789527971404013309218848694542736 (base 10) = 202222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222211 (base 3)
a(26) = 135682719673760139277577056 (base 10) = 2022222222222222222222222222222222222222222222222222211 (base 3)
a(27) = 3103985417701264389637747414334049249616 (base 10) = 20222222222222222222222222222222222222222222222222222222222222222222222222222222211 (base 3)
a(28) = 2262805369504221740045917865049521902973704 (base 10) = 20222222222222222222222222222222222222222222222222222222222222222222222222222222222222211 (base 3)
a(29) = 98912702642171141533353677464 (base 10) = 2022222222222222222222222222222222222222222222222222222222211 (base 3)
a(30) = 890214323779540273800183097216 (base 10) = 202222222222222222222222222222222222222222222222222222222222211 (base 3)
a(33) = 648966242035284859600333477874104 (base 10) = 202222222222222222222222222222222222222222222222222222222222222222211 (base 3)
a(34) = 1624463421305399727955317383718278234275809671395357663852488383536 (base 10) = 2022222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222211 (base 3)
a(35) = 52566265604858073627627011707802824 (base 10) = 2022222222222222222222222222222222222222222222222222222222222222222222211 (base 3)
a(36) = 14846266029317198836441267112589913205410504744 (base 10) = 2022222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222211 (base 3)
a(39) = 344887268633473821070860823814894361064 (base 10) = 202222222222222222222222222222222222222222222222222222222222222222222222222222211 (base 3)
a(40) = 1166199021234168600322186441674903634837253284669268066647739969989548815584492779042071704 (base 10) = 202222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222211 (base 3)
a(41) = 27935868759311379506739726729006443246584 (base 10) = 2022222222222222222222222222222222222222222222222222222222222222222222222222222222211 (base 3)
a(42) = 95922940564662548536033536191180611455752285286224474692825586559712504 (base 10) = 20222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222211 (base 3)
a(44) = 51765728804119417583918760520852820074569256760349620344 (base 10) = 202222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222211 (base 3)
a(46) = 131581537125737377964380708081180536976340583383023970772051559066816 (base 10) = 20222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222211 (base 3)
a(48) = 850159086479708909634873915981004749796357644523896420586202438122381086561095235921670275856 (base 10) = 202222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222211 (base 3)
a(50) = 10822927935372237951765683725078046726744257962016 (base 10) = 2022222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222211 (base 3)
a(51) = 97406351418350141565891153525702420540698321658184 (base 10) = 202222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222211 (base 3)
a(52) = 50977383456624829456538198506977215332656450244796421055229922546880175495464 (base 10) = 20222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222211 (base 3)
a(53) = 7889914464886361466837183435581896063796564054313304 (base 10) = 2022222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222211 (base 3)
a(55) = 37737216298203055418676776419701705834360988178294873234416 (base 10) = 202222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222211 (base 3)
a(60) = 1184233834131636401679426372730624832787065250447215736948464031601384 (base 10) = 2022222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222211 (base 3)
a(65) = 2228344885192592219417444970806966027813181990940133969619325624 (base 10) = 2022222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222211 (base 3)
a(69) = 14620170791748597551597856453464504108482287042558218974672395451864 (base 10) = 202222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222211 (base 3)
a(74) = 863306465081962936824301825720625503101770567576020272235430279037412576 (base 10) = 2022222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222211 (base 3)
a(78) = 3010163515730239554579124083638497588178030730504983866890271696470727482831948976 (base 10) = 202222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222211 (base 3)
a(81) = 4129168059986611185979594079065154441945172469828510105473623726297294215132984 (base 10) = 202222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222211 (base 3)
a(83) = 334462612858915506064347120404277509797558970056109318543363521830080831425772104 (base 10) = 2022222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222211 (base 3)
a(86) = 243823244774149403920909050774718304642420489170903693218112007414128926109387867456 (base 10) = 2022222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222211 (base 3)
a(89) = 177747145440354915458342698014769644084324536605588792356003653404899987133743755379064 (base 10) = 2022222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222211 (base 3)
a(90) = 1599724308963194239125084282132926796758920829450299131204032880644099884203693798411616 (base 10) = 202222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222211 (base 3)
a(95) = 94462120719967656626097101775667194421817516058210713398466937569153454062343915102407808424 (base 10) = 2022222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222211 (base 3)
a(98) = 68862886004856421680424787194461384733504969206435610067482397487912868011448714109655292344736 (base 10) = 2022222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222211 (base 3)

Index entries for the Kaprekar map

Cf. A164993, A164997, A164998, A165000, A165002, A165004.

In other bases: A153881 (base 2), A165028 (base 4), A165047 (base 5), A165067 (base 6), A165086 (base 7), A165106 (base 8), A165126 (base 9), A151959 (base 10).

A165028 Consider the base-4 Kaprekar map x->K(x) described in A165012. Sequence gives the smallest number that belongs to a cycle of length n under repeated iteration of this map, or -1 if there is no cycle of length n.

Original entry on oeis.org

0, 126, 41958

Offset: 1

Joseph Myers, Sep 04 2009

Known values (to 200 base-4 digits):
a(1) = 0 (base 10) = 0 (base 4)
a(2) = 126 (base 10) = 1332 (base 4)
a(3) = 41958 (base 10) = 22033212 (base 4)

Index entries for the Kaprekar map

Cf. A165012, A165016, A165017, A165019, A165021, A165023.

In other bases: A153881 (base 2), A165008 (base 3), A165047 (base 5), A165067 (base 6), A165086 (base 7), A165106 (base 8), A165126 (base 9), A151959 (base 10).

A165067 Consider the base-6 Kaprekar map x->K(x) described in A165051. Sequence gives the smallest number that belongs to a cycle of length n under repeated iteration of this map, or -1 if there is no cycle of length n.

Original entry on oeis.org

0, 4305, 16840

Offset: 1

Joseph Myers, Sep 04 2009

Known values (to 100 base-6 digits):
a(1) = 0 (base 10) = 0 (base 6)
a(2) = 4305 (base 10) = 31533 (base 6)
a(3) = 16840 (base 10) = 205544 (base 6)
a(6) = 430 (base 10) = 1554 (base 6)
a(7) = 895275 (base 10) = 31104443 (base 6)

Index entries for the Kaprekar map

Cf. A165051, A165055, A165056, A165058, A165060, A165062.

In other bases: A153881 (base 2), A165008 (base 3), A165028 (base 4), A165047 (base 5), A165086 (base 7), A165106 (base 8), A165126 (base 9), A151959 (base 10).

A165086 Consider the base-7 Kaprekar map x->K(x) described in A165071. Sequence gives the smallest number that belongs to a cycle of length n under repeated iteration of this map, or -1 if there is no cycle of length n.

Original entry on oeis.org

0, 144, 1068, 9458722410775248, 9936, 55500, 65945195409025452

Offset: 1

Joseph Myers, Sep 04 2009

Known values (to 100 base-7 digits):
a(1) = 0 (base 10) = 0 (base 7)
a(2) = 144 (base 10) = 264 (base 7)
a(3) = 1068 (base 10) = 3054 (base 7)
a(4) = 9458722410775248 (base 10) = 5544222066654442212 (base 7)
a(5) = 9936 (base 10) = 40653 (base 7)
a(6) = 55500 (base 10) = 320544 (base 7)
a(7) = 65945195409025452 (base 10) = 55332221066554443312 (base 7)
a(9) = 419850417612 (base 10) = 42222166444443 (base 7)
a(10) = 114965566537586468276798389479111631100827277423731225926928273344 (base 10) = 65444444444444444444444443066666666666666666666666532222222222222222222222211 (base 7)
a(11) = 31412208 (base 10) = 530666532 (base 7)
a(12) = 26884299308652 (base 10) = 5443216666443222 (base 7)
a(13) = 894060461610805641013834968 (base 10) = 54444444322106666665544322222222 (base 7)
a(14) = 1591271424672409468790707489057394638817384701224062547077367141620193382944 (base 10) = 65444444444444444444444444444306666666666666666666666666665322222222222222222222222222211 (base 7)
a(17) = 107837050564847832079804652808012 (base 10) = 55444444332221110666655554443322222212 (base 7)
a(24) = 7598644111289477155212 (base 10) = 54443222221066554444432222 (base 7)
a(25) = 18244344524504743400068812 (base 10) = 544432222222106655444444432222 (base 7)

Index entries for the Kaprekar map

Cf. A165071, A165075, A165076, A165078, A165080, A165082.

In other bases: A153881 (base 2), A165008 (base 3), A165028 (base 4), A165047 (base 5), A165067 (base 6), A165106 (base 8), A165126 (base 9), A151959 (base 10).

cp's OEIS Frontend

A151949 a(n) = image of n under the Kaprekar map n -> (n with digits sorted into descending order) - (n with digits sorted into ascending order).

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A099009 Fixed points of the Kaprekar mapping f(n) = n' - n'', where in n' the digits of n are arranged in descending, in n'' in ascending order.

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A165047 Consider the base-5 Kaprekar map x->K(x) described in A165032. Sequence gives the smallest number that belongs to a cycle of length n under repeated iteration of this map, or -1 if there is no cycle of length n.

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A165126 Consider the base-9 Kaprekar map x->K(x) described in A165110. Sequence gives the smallest number that belongs to a cycle of length n under repeated iteration of this map, or -1 if there is no cycle of length n.

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A165008 Consider the base-3 Kaprekar map x->K(x) described in A164993. Sequence gives the smallest number that belongs to a cycle of length n under repeated iteration of this map, or -1 if there is no cycle of length n.

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A165028 Consider the base-4 Kaprekar map x->K(x) described in A165012. Sequence gives the smallest number that belongs to a cycle of length n under repeated iteration of this map, or -1 if there is no cycle of length n.

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A165067 Consider the base-6 Kaprekar map x->K(x) described in A165051. Sequence gives the smallest number that belongs to a cycle of length n under repeated iteration of this map, or -1 if there is no cycle of length n.

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A165086 Consider the base-7 Kaprekar map x->K(x) described in A165071. Sequence gives the smallest number that belongs to a cycle of length n under repeated iteration of this map, or -1 if there is no cycle of length n.

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A165106 Consider the base-8 Kaprekar map x->K(x) described in A165090. Sequence gives the smallest number that belongs to a cycle of length n under repeated iteration of this map, or -1 if there is no cycle of length n.

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A164723 Numbers belonging to cycles of length 2 under the Kaprekar map A151949.

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