A099010 -id:A099010 - cp's OEIS frontend

A165078 Consider the base-7 Kaprekar map n->K(n) defined in A165071. Sequence gives numbers belonging to cycles of length greater than 1.

144, 192, 1068, 1752, 1836, 9936, 10608, 13008, 13680, 15072, 55500, 72012, 77388, 89112, 91212, 91800, 640992, 643344, 660144, 674592, 760992, 780144, 3562968, 4605036, 4642668, 4772952, 5445336, 5464152, 31412208, 31680336, 32353344

Offset: 1

Joseph Myers, Sep 04 2009

Initial terms in base 7: 264, 363, 3054, 5052, 5232, 40653, 42633, 52632, 54612, 61641.

Joseph Myers, Table of n, a(n) for n=1..32041
Index entries for the Kaprekar map

Cf. A165071, A165079, A165075, A165076, A165080, A165082, A165088.

In other bases: Empty (base 2), A165000 (base 3), A165019 (base 4), A165039 (base 5), A165058 (base 6), A165097 (base 8), A165117 (base 9), A099010 (base 10).

A165097 Consider the base-8 Kaprekar map n->K(n) defined in A165090. Sequence gives numbers belonging to cycles of length greater than 1.

Original entry on oeis.org

1022, 1589, 2044, 2212, 2723, 3122, 3178, 3290, 17892, 20475, 21483, 21987, 25578, 26586, 102837, 177443, 217938, 1445787, 1449819, 1646442, 1707930, 1715994, 1740690, 1752786, 1777482, 1941345, 1978137, 1982169, 6589877, 11381027

Offset: 1

Joseph Myers, Sep 04 2009

Initial terms in base 8: 1776, 3065, 3774, 4244, 5243, 6062, 6152, 6332, 42744, 47773.

Joseph Myers, Table of n, a(n) for n=1..25468
Index entries for the Kaprekar map

Cf. A165090, A165098, A165094, A165095, A165099, A165101, A165108.

In other bases: Empty (base 2), A165000 (base 3), A165019 (base 4), A165039 (base 5), A165058 (base 6), A165078 (base 7), A165117 (base 9), A099010 (base 10).

A165117 Consider the base-9 Kaprekar map n->K(n) defined in A165110. Sequence gives numbers belonging to cycles of length greater than 1.

Original entry on oeis.org

16, 48, 320, 400, 2256, 3712, 3856, 5168, 5312, 5456, 34960, 40080, 42240, 49520, 55360, 183696, 250592, 308992, 322096, 323392, 388992, 419888, 440192, 446096, 448688, 454592, 505792, 511696, 512992, 3496800, 3916640, 31596672, 35193008

Offset: 1

Joseph Myers, Sep 04 2009

Initial terms in base 9: 17, 53, 385, 484, 3076, 5074, 5254, 7072, 7252, 7432.

Joseph Myers, Table of n, a(n) for n=1..7548
Index entries for the Kaprekar map

Cf. A165110, A165118, A165114, A165115, A165119, A165121, A165128.

In other bases: Empty (base 2), A165000 (base 3), A165019 (base 4), A165039 (base 5), A165058 (base 6), A165078 (base 7), A165097 (base 8), A099010 (base 10).

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

Original entry on oeis.org

53955, 59994, 8733209876622, 9665429654331, 873332098766622, 966543296654331, 8764421997755322, 8765431997654322, 87333320987666622, 96654332966654331, 8733333209876666622, 9665433329666654331

Offset: 1

Joseph Myers, Aug 23 2009

Joseph Myers, Table of n, a(n) for n=1..1146
Index entries for the Kaprekar map

Cf. A151949, A151959, A099009, A099010, A164716, A164725, A164727, A164729, A164724.

A164725 Numbers belonging to cycles of length 3 under the Kaprekar map A151949.

Original entry on oeis.org

64308654, 83208762, 86526432, 6431088654, 6433086654, 6543086544, 8321088762, 8332087662, 8653266432, 8655264432, 8732087622, 8765264322, 9751088421, 9755084421, 9775084221, 643110888654, 643310886654, 643330866654

Offset: 1

Joseph Myers, Aug 23 2009

Joseph Myers, Table of n, a(n) for n=1..27849
Index entries for the Kaprekar map

Cf. A151949, A151959, A099009, A099010, A164716, A164723, A164727, A164729, A164726.

A164727 Numbers belonging to cycles of length 5 under the Kaprekar map A151949.

Original entry on oeis.org

86420987532, 86541975432, 87641975322, 88431976512, 96641975331, 8643209876532, 8654209875432, 8654319765432, 8764209875322, 8764319765322, 8765419754322, 8843209876512, 8843319766512, 8854319765412, 8874319765212

Offset: 1

Joseph Myers, Aug 23 2009

Joseph Myers, Table of n, a(n) for n=1..14510
Index entries for the Kaprekar map

Cf. A151949, A151959, A099009, A099010, A164716, A164723, A164725, A164729, A164728.

A164729 Numbers belonging to cycles of length 7 under the Kaprekar map A151949.

Original entry on oeis.org

420876, 642654, 750843, 840852, 851742, 860832, 862632, 43208766, 64326654, 75308643, 84308652, 85317642, 86308632, 86326632, 4332087666, 6433266654, 7533086643, 8433086652, 8533176642, 8633086632, 8633266632, 433320876666

Offset: 1

Joseph Myers, Aug 23 2009

Joseph Myers, Table of n, a(n) for n=1..476
Index entries for the Kaprekar map

Cf. A151949, A151959, A099009, A099010, A164716, A164723, A164725, A164727, A164730.

A160761 The Kaprekar binary numbers in decimal.

Original entry on oeis.org

9, 9, 9, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 45, 45, 49, 45, 49, 49, 45, 45, 49, 49, 45, 49, 45, 45, 45, 49, 49, 45, 49, 45, 45, 49, 45, 45, 45, 93, 93, 105, 93, 105, 105, 105, 93, 105, 105, 105, 105, 105, 105, 93, 93, 105, 105, 105, 105, 105, 105, 93, 105, 105, 105

Offset: 1

Damir Olejar, May 25 2009

			The number 9 is 1001 in binary. The maximum number using the same number of 0's and one's is found and the minimum number having the same number of 0's and 1's is found to obtain the equation such as 1100 - 0011 = 1001. Repeating the same procedure always gives us the same number and pattern of 0's and 1's. Therefore 9 is one of the Kaprekar numbers. Numbers that end the procedure in 0 are excluded since they are not Kaprekar numbers.

M. Charosh, Some Applications of Casting Out 999...'s, Journal of Recreational Mathematics 14, 1981-82, pp. 111-118
D. R. Kaprekar, On Kaprekar numbers, J. Rec. Math., 13 (1980-1981), 81-82.

Cf. A099009, A099010, A090429, A069746, A163205.

class pattern { public static void main(String args[]) { int mem1 = 0; int mem2 =1; for (int i = 1; i<3000; i++) {do { mem1 = mem2; String binaryi = Integer.toBinaryString(i); String binarysort = ""; String binaryminimum = ""; for (int n = 0; n< binaryi.length(); n++) { String g = binaryi.substring(n,n+1); if (g.equals("0")){ binarysort = binarysort+"0"; } else { binarysort = "1"+binarysort; binaryminimum = binaryminimum + "1"; } } int binrev1 = Integer.parseInt(binarysort , 2); int binrev2 = Integer.parseInt(binaryminimum , 2); int diff = binrev1 - binrev2; mem2 = diff; } while (mem2!=0 && mem2!=mem1); String memtobin = Integer.toBinaryString(mem1); int ones = 0; for (int t = 0; t

nmax = 100; f[n_] := Module[{id, sid, min, max}, id = IntegerDigits[n, 2]; min = FromDigits[sid = Sort[id], 2]; max = FromDigits[Reverse[sid], 2]; max - min]; Reap[Do[If[(fpn = FixedPoint[f, n]) > 0, Sow[fpn]], {n, 1, nmax}]][[2, 1]] (* Jean-François Alcover, Apr 23 2017 *)

1. Sort all integers from the number in descending order 2. Sort all integers from the number in ascending order 3. Subtract ascending from descending order to obtain a new number 4. Repeat the steps 1-3 with a new number until a repetitive sequence is obtained or until a zero is obtained. 5. Call the repetitive sequence's number a Kaprekar number, ignore zeros.

A177686 If a1a2a3 is a 3-digit integer in a concatenated form, we define two permutations of its digits as follows: P1(a1a2a3)=a2a3a1 and P2(a1a2a3)=a1a3a2, then we take the absolute value of their difference. Thus we form a sequence: a1a2a3, abs(P1(a1a2a3)-P2(a1a2a3)), and so on.

Original entry on oeis.org

99, 891, 198, 792, 297, 693, 396, 594, 495

Offset: 1

F. Smarandache (smarand(AT)unm.edu), May 10 2010

This is an alternative to Kaprekar's routine. It would be interested in studying 4-digit integers with the permutations P1(a1a2a3a4)=a2a3a4a1 and P2(a1a2a3a4)=a1a3a4a2. Other permutations can be also studied. A generalization of Kaprekar's routine is the following: Let f be an operator that maps a finite set A={a1, a2, ..., a_p}, with p>=1 elements, into itself. Then, for any value 'a' in A, we have f(a) belongs to A too. If we iterate this operator multiple times, we get a chain: a, f(a), f(f(a)), ..., f(f...f(a)...), ... all of whose elements are in A. But, since A is finite, after at most p iterations we get two equal iterations. Therefore we end up in a finite cycle (of one or more terms).

			Starting with 100, we get abs(001-100)=099, then abs(990-099)=891, then abs(918-819)=099, etc. So 100, 099, 891, 099, ... (the cycle is 099, 891). Each three-digit number ends up in a cycle of two terms (such as: 99 and 891, or 198 and 792, or 297 and 693, or 396 and 594), or in a constant 495 (as in Kaprekar's routine).
Starting with 495, we get abs(954-459)=495 (cycle of one term).

F. Smarandache, Proposed Problems of Mathematics, Vol. II, State University of Moldova Press, Kishinev, pp. 83-84, 1997.
F. Smarandache, Generalization and alternatives of Kaprekar's routine, Multispace & Multistructure / Neutrosophic Transdisciplinarity, Northern-European Publishers, pp. 555-559, Finland, 2010. arXiv:1005.3235

Cf. A090429, A099010.

abs(P1(a1a2a3)-P2(a1a2a3)) = abs(a2a3a1-a1a3a2) = 99x(a2-a1).

Added keyword:base,fini,full as there are only 9 different values obtained by the abs() starting from any a1a2a3 in the range 100 to 999 R. J. Mathar, May 15 2010

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A165078 Consider the base-7 Kaprekar map n->K(n) defined in A165071. Sequence gives numbers belonging to cycles of length greater than 1.

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A165097 Consider the base-8 Kaprekar map n->K(n) defined in A165090. Sequence gives numbers belonging to cycles of length greater than 1.

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A165117 Consider the base-9 Kaprekar map n->K(n) defined in A165110. Sequence gives numbers belonging to cycles of length greater than 1.

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

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A164725 Numbers belonging to cycles of length 3 under the Kaprekar map A151949.

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A164727 Numbers belonging to cycles of length 5 under the Kaprekar map A151949.

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A164729 Numbers belonging to cycles of length 7 under the Kaprekar map A151949.

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A160761 The Kaprekar binary numbers in decimal.

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A177686 If a1a2a3 is a 3-digit integer in a concatenated form, we define two permutations of its digits as follows: P1(a1a2a3)=a2a3a1 and P2(a1a2a3)=a1a3a2, then we take the absolute value of their difference. Thus we form a sequence: a1a2a3, abs(P1(a1a2a3)-P2(a1a2a3)), and so on.

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