A165003 -id:A165003 - cp's OEIS frontend

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

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).

A164719 Length of cycle mentioned in A164718.

Original entry on oeis.org

1, 1, 1, 2, 4, 4, 7, 1, 1, 8, 7, 1, 3, 1, 1, 14, 1, 7, 1, 3, 3, 3, 3, 1, 1, 8, 5, 1, 7, 1, 1, 3, 3, 3, 3, 3, 3, 3, 3, 1, 3, 3, 1, 1, 5, 1, 5, 2, 5, 7, 1, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 3, 3, 3, 1, 3, 3, 1, 3, 3, 1, 1, 1, 5, 1, 5, 2, 5, 5, 5, 7, 1, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2, 3, 3, 3

Offset: 1

Joseph Myers, Aug 23 2009

Joseph Myers, Table of n, a(n) for n=1..10163
Index entries for the Kaprekar map

Cf. A151949, A164718, A164715, A164717, A164721, A151966.

In other bases: A000012 (base 2), A165003 (base 3), A165022 (base 4), A165042 (base 5), A165061 (base 6), A165081 (base 7), A165100 (base 8), A165120 (base 9). [From Joseph Myers, Sep 05 2009]

A164999 Length of cycle mentioned in A164998.

Original entry on oeis.org

1, 2, 2, 1, 3, 3, 3, 2, 2, 1, 1, 3, 3, 3, 2, 2, 1, 4, 4, 4, 4, 1, 3, 3, 1, 3, 3, 3, 3, 2, 2, 1, 2, 2, 1, 5, 5, 5, 5, 5, 3, 3, 1, 3, 3, 3, 3, 2, 2, 1, 1, 2, 2, 1, 6, 6, 6, 6, 6, 6, 5, 5, 5, 5, 5, 3, 3, 1, 3, 3, 3, 3, 2, 2, 1, 3, 3, 3, 1, 2, 2, 1, 4, 4, 2, 4, 1, 2, 4, 6, 6, 6, 6, 6, 6, 5, 5, 5, 5, 5, 3, 3, 1, 3, 3

Offset: 1

Joseph Myers, Sep 04 2009

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

Cf. A164993, A164998, A165001, A165003, A165005, A165011.

In other bases: A000012 (base 2), A165018 (base 4), A165038 (base 5), A165057 (base 6), A165077 (base 7), A165096 (base 8), A165116 (base 9), A164717 (base 10).

A165001 Length of cycle mentioned in A165000.

Original entry on oeis.org

2, 2, 3, 3, 3, 2, 2, 3, 3, 3, 2, 2, 4, 4, 4, 4, 3, 3, 3, 3, 3, 3, 2, 2, 2, 2, 5, 5, 5, 5, 5, 3, 3, 3, 3, 3, 3, 2, 2, 2, 2, 6, 6, 6, 6, 6, 6, 5, 5, 5, 5, 5, 3, 3, 3, 3, 3, 3, 2, 2, 3, 3, 3, 2, 2, 4, 4, 2, 4, 2, 4, 6, 6, 6, 6, 6, 6, 5, 5, 5, 5, 5, 3, 3, 3, 3, 3, 3, 2, 2, 5, 5, 5, 5, 5, 3, 3, 3, 2, 2, 4, 4, 4, 4, 4

Offset: 1

Joseph Myers, Sep 04 2009

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

Cf. A164993, A165000, A164999, A165003, A165005, A165011.

In other bases: Empty (base 2), A165020 (base 4), A165040 (base 5), A165059 (base 6), A165079 (base 7), A165098 (base 8), A165118 (base 9), A164715 (base 10).

A165005 Length of cycle mentioned in A165004.

Original entry on oeis.org

2, 3, 2, 3, 2, 4, 3, 3, 2, 2, 5, 3, 3, 2, 2, 6, 5, 3, 3, 2, 3, 2, 4, 2, 6, 5, 3, 3, 2, 5, 3, 2, 4, 4, 4, 2, 6, 5, 3, 3, 2, 3, 3, 2, 9, 4, 4, 4, 2, 6, 5, 3, 3, 2, 2, 3, 3, 2, 6, 3, 9, 4, 4, 4, 2, 6, 5, 3, 3, 2, 5, 2, 3, 3, 2, 11, 6, 3, 9, 4, 4, 4, 2, 6, 5, 3, 3, 2, 5, 2, 3, 3, 2, 10, 2, 11, 6, 3, 9, 4, 4, 4, 2, 6

Offset: 1

Joseph Myers, Sep 04 2009

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

Cf. A164993, A165004, A164999, A165001, A165003, A165011.

In other bases: Empty (base 2), A165024 (base 4), A165044 (base 5), A165063 (base 6), A165083 (base 7), A165102 (base 8), A165122 (base 9), A164721 (base 10).

A165022 Length of cycle mentioned in A165021.

Original entry on oeis.org

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

Offset: 1

Joseph Myers, Sep 04 2009

Joseph Myers, Table of n, a(n) for n=1..7165
Anthony Kay and Katrina Downes-Ward, Fixed Points and Cycles of the Kaprekar Transformation: 2. Even bases, arXiv:2408.12257 [math.CO], 2024. See p. 16.
Index entries for the Kaprekar map

Cf. A165012, A165021, A165018, A165020, A165024, A165031.

In other bases: A000012 (base 2), A165003 (base 3), A165042 (base 5), A165061 (base 6), A165081 (base 7), A165100 (base 8), A165120 (base 9), A164719 (base 10).

A165042 Length of cycle mentioned in A165041.

Original entry on oeis.org

1, 1, 2, 1, 4, 5, 4, 6, 1, 4, 3, 1, 8, 5, 3, 1, 5, 4, 4, 6, 2, 5, 3, 1, 5, 4, 1, 4, 2, 2, 4, 6, 2, 5, 3, 1, 5, 1, 3, 4, 4, 4, 4, 4, 2, 2, 4, 6, 2, 5, 3, 1, 3, 1, 9, 9, 3, 4, 4, 4, 4, 4, 2, 2, 4, 6, 2, 5, 3, 1, 5, 5, 3, 1, 6, 6, 2, 6, 3, 3, 1, 9, 9, 3, 4, 4, 4, 4, 4, 2, 2, 4, 6, 2, 5, 3, 1, 11, 5, 3, 1, 11, 11

Offset: 1

Joseph Myers, Sep 04 2009

Joseph Myers, Table of n, a(n) for n=1..15716
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

Cf. A165032, A165041, A165038, A165040, A165044, A165050.

In other bases: A000012 (base 2), A165003 (base 3), A165022 (base 4), A165061 (base 6), A165081 (base 7), A165100 (base 8), A165120 (base 9), A164719 (base 10).

A165061 Length of cycle mentioned in A165060.

Original entry on oeis.org

1, 1, 6, 2, 1, 3, 1, 1, 1, 2, 7, 2, 1, 2, 1, 1, 2, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 2, 2, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 2, 1, 2, 2, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 1, 1, 2, 2, 2, 2, 1, 2, 1, 1, 2, 2, 2, 1, 1, 2, 2, 2, 1, 2, 2, 2, 2, 1, 2, 2, 1, 2, 2

Offset: 1

Joseph Myers, Sep 04 2009

Joseph Myers, Table of n, a(n) for n=1..23045
Anthony Kay and Katrina Downes-Ward, Fixed Points and Cycles of the Kaprekar Transformation: 2. Even bases, arXiv:2408.12257 [math.CO], 2024. See p. 27.
Index entries for the Kaprekar map

Cf. A165051, A165060, A165057, A165059, A165063, A165070.

In other bases: A000012 (base 2), A165003 (base 3), A165022 (base 4), A165042 (base 5), A165081 (base 7), A165100 (base 8), A165120 (base 9), A164719 (base 10).

A165081 Length of cycle mentioned in A165080.

Original entry on oeis.org

1, 2, 3, 5, 6, 6, 6, 11, 2, 2, 2, 1, 2, 5, 6, 1, 1, 9, 2, 2, 1, 1, 1, 2, 12, 1, 2, 1, 1, 1, 2, 4, 1, 2, 1, 2, 1, 1, 1, 2, 5, 7, 2, 2, 1, 1, 2, 3, 1, 2, 1, 2, 1, 1, 1, 2, 5, 1, 2, 1, 1, 2, 3, 1, 2, 3, 1, 2, 1, 2, 1, 1, 1, 2, 5, 5, 1, 1, 2, 1, 2, 2, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 1, 2, 1, 1, 1, 2, 24, 1, 1, 1, 2

Offset: 1

Joseph Myers, Sep 04 2009

Joseph Myers, Table of n, a(n) for n=1..13070
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

Cf. A165071, A165080, A165077, A165079, A165083, A165089.

In other bases: A000012 (base 2), A165003 (base 3), A165022 (base 4), A165042 (base 5), A165061 (base 6), A165100 (base 8), A165120 (base 9), A164719 (base 10).

A165100 Length of cycle mentioned in A165099.

Original entry on oeis.org

1, 1, 1, 5, 3, 2, 4, 3, 1, 1, 7, 4, 1, 3, 3, 1, 1, 5, 5, 4, 3, 3, 3, 3, 1, 6, 4, 2, 3, 1, 3, 3, 3, 4, 1, 2, 1, 3, 3, 3, 1, 5, 9, 3, 3, 3, 3, 3, 3, 1, 1, 3, 3, 3, 1, 3, 3, 3, 1, 1, 12, 1, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 3, 3, 3, 1, 2, 5, 1, 3, 1, 3, 1, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 6, 2, 3, 3, 3

Offset: 1

Joseph Myers, Sep 04 2009

Joseph Myers, Table of n, a(n) for n=1..8775
Anthony Kay and Katrina Downes-Ward, Fixed Points and Cycles of the Kaprekar Transformation: 2. Even bases, arXiv:2408.12257 [math.CO], 2024. See p. 33.
Index entries for the Kaprekar map

Cf. A165090, A165099, A165096, A165098, A165102, A165109.

In other bases: A000012 (base 2), A165003 (base 3), A165022 (base 4), A165042 (base 5), A165061 (base 6), A165081 (base 7), A165120 (base 9), A164719 (base 10).

cp's OEIS Frontend

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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A164719 Length of cycle mentioned in A164718.

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A164999 Length of cycle mentioned in A164998.

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A165001 Length of cycle mentioned in A165000.

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A165005 Length of cycle mentioned in A165004.

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A165022 Length of cycle mentioned in A165021.

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A165042 Length of cycle mentioned in A165041.

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A165061 Length of cycle mentioned in A165060.

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A165081 Length of cycle mentioned in A165080.

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A165100 Length of cycle mentioned in A165099.

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