A164993 -id:A164993 - cp's OEIS frontend

A164998 Consider the base-3 Kaprekar map n->K(n) defined in A164993. Sequence gives numbers belonging to cycles, including fixed points.

0, 32, 52, 184, 320, 484, 580, 1696, 1768, 2008, 5332, 15304, 16024, 16240, 18208, 18424, 19144, 26240, 39364, 48100, 48964, 55360, 137776, 144328, 146488, 147136, 164008, 166168, 166816, 172720, 173368, 175528, 433012, 441652, 520372, 1240024

Offset: 1

Joseph Myers, Sep 04 2009

Initial terms in base 3: 0, 1012, 1221, 20211, 102212, 122221, 210111, 2022211, 2102111, 2202101.

Joseph Myers, Table of n, a(n) for n=1..8662
Index entries for the Kaprekar map

Union of A164997 and A165000. Cf. A164993, A164999, A165002, A165004, A165010, A165007.

In other bases: A163205 (base 2), A165017 (base 4), A165037 (base 5), A165056 (base 6), A165076 (base 7), A165095 (base 8), A165115 (base 9), A164716 (base 10).

A164997 List of fixed points of the base-3 Kaprekar map A164993.

Original entry on oeis.org

0, 184, 2008, 5332, 19144, 55360, 146488, 175528, 520372, 1502344, 1589464, 3975844, 4749760, 14072968, 14334328, 40625344, 42947092, 107527864, 128312344, 129096424, 380156356, 387121600, 1097424568, 1159778008

Offset: 1

Joseph Myers, Sep 04 2009

Initial terms in base 3: 0, 20211, 2202101, 21022111, 222021001, 2210221101, 21102221111, 22220210001, 222102211001, 2211022211101.

Joseph Myers, Table of n, a(n) for n=1..3268
Index entries for the Kaprekar map

Cf. A164993, A164998, A165000, A008615.

In other bases: A163205 (base 2), A165016 (base 4), A165036 (base 5), A165055 (base 6), A165075 (base 7), A165094 (base 8), A165114 (base 9), A099009 (base 10).

A165000 Consider the base-3 Kaprekar map n->K(n) defined in A164993. Sequence gives numbers belonging to cycles of length greater than 1.

Original entry on oeis.org

32, 52, 320, 484, 580, 1696, 1768, 15304, 16024, 16240, 18208, 18424, 26240, 39364, 48100, 48964, 137776, 144328, 147136, 164008, 166168, 166816, 172720, 173368, 433012, 441652, 1240024, 1299064, 1318720, 1325200, 1327144, 1476208, 1495864

Offset: 1

Joseph Myers, Sep 04 2009

Initial terms in base 3: 1012, 1221, 102212, 122221, 210111, 2022211, 2102111, 202222211, 210222111, 211021111.

Joseph Myers, Table of n, a(n) for n=1..8276
Index entries for the Kaprekar map

Cf. A164993, A165001, A164997, A164998, A165002, A165004, A165010.

In other bases: Empty (base 2), A165019 (base 4), A165039 (base 5), A165058 (base 6), A165078 (base 7), A165097 (base 8), A165117 (base 9), A099010 (base 10).

A165002 Consider the base-3 Kaprekar map n->K(n) defined in A164993. Sequence gives least elements of each cycle, including fixed points.

Original entry on oeis.org

0, 32, 184, 320, 1696, 2008, 5332, 15304, 18208, 19144, 26240, 55360, 137776, 146488, 164008, 172720, 175528, 433012, 520372, 1240024, 1476208, 1502344, 1554904, 1581040, 1589464, 3975844, 4487680, 4749760, 11160256, 13286008

Offset: 1

Joseph Myers, Sep 04 2009

Initial terms in base 3: 0, 1012, 20211, 102212, 2022211, 2202101, 21022111, 202222211, 220222101, 222021001.

Joseph Myers, Table of n, a(n) for n=1..23340
Anthony Kay and Katrina Downes-Ward, Fixed Points and Cycles of the Kaprekar Transformation: 1. Odd Bases, Journal of Integer Sequences, Vol. 25 (2022), Article 22.6.7.
Index entries for the Kaprekar map

Union of A164997 and A165004. Cf. A164993, A165003, A164998, A165000, A165010, A165006.

In other bases: A163205 (base 2), A165021 (base 4), A165041 (base 5), A165060 (base 6), A165080 (base 7), A165099 (base 8), A165119 (base 9), A164718 (base 10).

A165004 Consider the base-3 Kaprekar map n->K(n) defined in A164993. Sequence gives least elements of each cycle of length > 1.

Original entry on oeis.org

32, 320, 1696, 15304, 18208, 26240, 137776, 164008, 172720, 433012, 1240024, 1476208, 1554904, 1581040, 4487680, 11160256, 13286008, 13994560, 14230648, 14309056, 35075092, 42160852, 100442344, 106819600, 119574208, 125951464

Offset: 1

Joseph Myers, Sep 04 2009

Initial terms in base 3: 1012, 102212, 2022211, 202222211, 220222101, 1022222212, 20222222211, 22022222101, 22202221001, 210222222111.

Joseph Myers, Table of n, a(n) for n=1..20072
Index entries for the Kaprekar map

Cf. A164993, A165005, A164997, A164998, A165002, A165000, A165010.

In other bases: Empty (base 2), A165023 (base 4), A165043 (base 5), A165062 (base 6), A165082 (base 7), A165101 (base 8), A165121 (base 9), A164720 (base 10).

A165009 a(n) = smallest number that leads to a new cycle under the base-3 Kaprekar map of A164993.

Original entry on oeis.org

0, 28, 82, 244, 730, 737, 2188, 6562, 6569, 6587, 19684, 19691, 59050, 59057, 59062, 59075, 59129, 177148, 177173, 531442, 531449, 531467, 531485, 531521, 531683, 1594324, 1594331, 1594403, 4782970, 4782977, 4782995, 4783049, 4783103

Offset: 1

Joseph Myers, Sep 04 2009

Initial terms in base 3: 0, 1001, 10001, 100001, 1000001, 1000022, 10000001, 100000001, 100000022, 100000222.

Joseph Myers, Table of n, a(n) for n=1..23340
Index entries for the Kaprekar map

Cf. A164993, A164998, A165010, A165011.

In other bases: A164887 (base 2), A165029 (base 4), A165048 (base 5), A165068 (base 6), A165087 (base 7), A165107 (base 8), A165127 (base 9), A151964 (base 10).

A164995 Length of preperiodic part of trajectory of n under iteration of the base-3 Kaprekar map in A164993.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 2, 2, 1, 2, 2, 3, 2, 1, 2, 3, 2, 2, 3, 3, 3, 3, 2, 2, 3, 2, 1, 2, 1, 2, 1, 2, 0, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 0, 2, 1, 2, 2, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 1, 1, 2, 2, 2, 1, 2, 2, 3, 1, 3, 1, 1, 1, 2, 1, 3, 1

Offset: 0

Joseph Myers, Sep 04 2009

Joseph Myers, Table of n, a(n) for n=0..2187
Index entries for the Kaprekar map

Cf. A164993, A164996.

In other bases: A164885 (base 2), A165014 (base 4), A165034 (base 5), A165053 (base 6), A165073 (base 7), A165092 (base 8), A165112 (base 9), A151962 (base 10).

A164996 (Length of preperiodic part) + (length of cycle) of trajectory of n under iteration of the base-3 Kaprekar map in A164993.

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 3, 3, 2, 3, 3, 4, 3, 2, 3, 4, 3, 3, 4, 4, 4, 4, 3, 3, 4, 3, 2, 3, 3, 4, 3, 3, 2, 4, 3, 4, 3, 3, 3, 3, 2, 3, 3, 3, 3, 4, 3, 4, 3, 3, 3, 4, 2, 3, 3, 4, 4, 4, 3, 4, 4, 4, 3, 4, 3, 4, 3, 3, 3, 4, 3, 3, 4, 4, 3, 4, 3, 3, 3, 3, 2, 3, 3, 2, 3, 3, 2, 2, 2, 3, 3, 3, 2, 3, 3, 4, 2, 4, 2, 2, 2, 3, 2, 4, 2

Offset: 0

Joseph Myers, Sep 04 2009

Joseph Myers, Table of n, a(n) for n=0..2187
Index entries for the Kaprekar map

Cf. A164993, A164995.

In other bases: A164886 (base 2), A165015 (base 4), A165035 (base 5), A165054 (base 6), A165074 (base 7), A165093 (base 8), A165113 (base 9), A151963 (base 10).

A165006 Number of cycles of n-digit numbers (including fixed points) under the base-3 Kaprekar map A164993.

Original entry on oeis.org

1, 0, 0, 1, 1, 1, 2, 1, 3, 2, 5, 2, 6, 3, 7, 4, 10, 5, 12, 6, 13, 7, 16, 8, 17, 9, 19, 10, 22, 11, 23, 14, 26, 15, 30, 16, 33, 18, 34, 19, 37, 20, 39, 23, 42, 24, 47, 25, 48, 26, 50, 28, 55, 29, 56, 32, 59, 33, 63, 34, 64, 37, 65, 40, 72, 41, 78, 44, 79, 46, 82, 49, 83, 51, 87, 52

Offset: 1

Joseph Myers, Sep 04 2009

Joseph Myers, Table of n, a(n) for n=1..200
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.
Index entries for the Kaprekar map

Cf. A164993, A164998, A165007, A008615.

In other bases: A004526 (base 2, adjusted to start 1, 0, 0, 1, 1, ...), A165025 (base 4), A165045 (base 5), A165064 (base 6), A165084 (base 7), A165103 (base 8), A165123 (base 9), A164731 (base 10).

A165007 Number of n-digit numbers in a cycle (including fixed points) under the base-3 Kaprekar map A164993.

Original entry on oeis.org

1, 0, 0, 2, 1, 3, 3, 1, 6, 5, 10, 3, 15, 4, 21, 7, 28, 12, 36, 11, 45, 13, 55, 18, 66, 19, 78, 25, 91, 28, 105, 35, 120, 41, 136, 43, 153, 47, 171, 56, 190, 58, 210, 68, 231, 73, 253, 84, 276, 85, 300, 97, 325, 103, 351, 116, 378, 119, 406, 133, 435, 140, 465, 155, 496, 162

Offset: 1

Joseph Myers, Sep 04 2009

Joseph Myers, Table of n, a(n) for n=1..200
Index entries for the Kaprekar map

Cf. A164993, A164998, A165006, A008615.

In other bases: A004526 (base 2, adjusted to start 1, 0, 0, 1, 1, ...), A165026 (base 4), A165046 (base 5), A165065 (base 6), A165085 (base 7), A165104 (base 8), A165124 (base 9), A164732 (base 10).

cp's OEIS Frontend

A164998 Consider the base-3 Kaprekar map n->K(n) defined in A164993. Sequence gives numbers belonging to cycles, including fixed points.

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A164997 List of fixed points of the base-3 Kaprekar map A164993.

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A165000 Consider the base-3 Kaprekar map n->K(n) defined in A164993. Sequence gives numbers belonging to cycles of length greater than 1.

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A165002 Consider the base-3 Kaprekar map n->K(n) defined in A164993. Sequence gives least elements of each cycle, including fixed points.

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A165004 Consider the base-3 Kaprekar map n->K(n) defined in A164993. Sequence gives least elements of each cycle of length > 1.

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A165009 a(n) = smallest number that leads to a new cycle under the base-3 Kaprekar map of A164993.

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A164995 Length of preperiodic part of trajectory of n under iteration of the base-3 Kaprekar map in A164993.

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A164996 (Length of preperiodic part) + (length of cycle) of trajectory of n under iteration of the base-3 Kaprekar map in A164993.

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A165006 Number of cycles of n-digit numbers (including fixed points) under the base-3 Kaprekar map A164993.

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A165007 Number of n-digit numbers in a cycle (including fixed points) under the base-3 Kaprekar map A164993.

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Author

Keywords

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