A165081 -id:A165081 - cp's OEIS frontend

A165080 Consider the base-7 Kaprekar map n->K(n) defined in A165071. Sequence gives least elements of each cycle, including fixed points.

0, 144, 1068, 9936, 55500, 640992, 3562968, 31412208, 220709400, 227429400, 228238488, 1922263344, 11150046252, 11432420652, 75796404672, 94197649008, 96503566608, 419850417612, 546394287000, 3939440152944, 4615731883344

Offset: 1

Joseph Myers, Sep 04 2009

Initial terms in base 7: 0, 264, 3054, 40653, 320544, 5306532, 42166443, 530666532, 5316666432, 5431055322.

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

Union of A165075 and A165082. Cf. A165071, A165081, A165076, A165078, A165088, A165084.

In other bases: A163205 (base 2), A165002 (base 3), A165021 (base 4), A165041 (base 5), A165060 (base 6), 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]

A165003 Length of cycle mentioned in A165002.

Original entry on oeis.org

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

Offset: 1

Joseph Myers, Sep 04 2009

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

Cf. A164993, A165002, A164999, A165001, A165005, A165011.

In other bases: A000012 (base 2), A165022 (base 4), A165042 (base 5), A165061 (base 6), A165081 (base 7), A165100 (base 8), A165120 (base 9), A164719 (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).

A165077 Length of cycle mentioned in A165076.

Original entry on oeis.org

1, 2, 2, 3, 3, 3, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 2, 2, 2, 2, 2, 2, 1, 2, 2, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 1, 6, 1, 9, 9, 9, 2, 2, 9, 9, 9, 9, 9, 9, 2, 1, 2, 1, 1, 2, 2, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 1, 2

Offset: 1

Joseph Myers, Sep 04 2009

Joseph Myers, Table of n, a(n) for n=1..36122
Index entries for the Kaprekar map

Cf. A165071, A165076, A165079, A165081, A165083, A165089.

In other bases: A000012 (base 2), A164999 (base 3), A165018 (base 4), A165038 (base 5), A165057 (base 6), A165096 (base 8), A165116 (base 9), A164717 (base 10).

A165079 Length of cycle mentioned in A165078.

Original entry on oeis.org

2, 2, 3, 3, 3, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 2, 2, 2, 2, 2, 2, 2, 2, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 9, 9, 9, 2, 2, 9, 9, 9, 9, 9, 9, 2, 2, 2, 2, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 2, 2, 2, 2, 4, 4, 4, 2, 2

Offset: 1

Joseph Myers, Sep 04 2009

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

Cf. A165071, A165078, A165077, A165081, A165083, A165089.

In other bases: Empty (base 2), A165001 (base 3), A165020 (base 4), A165040 (base 5), A165059 (base 6), A165098 (base 8), A165118 (base 9), A164715 (base 10).

A165083 Length of cycle mentioned in A165082.

Original entry on oeis.org

2, 3, 5, 6, 6, 6, 11, 2, 2, 2, 2, 5, 6, 9, 2, 2, 2, 12, 2, 2, 4, 2, 2, 2, 5, 7, 2, 2, 2, 3, 2, 2, 2, 5, 2, 2, 3, 2, 3, 2, 2, 2, 5, 5, 2, 2, 2, 2, 3, 2, 3, 2, 3, 2, 2, 2, 24, 2, 2, 3, 2, 2, 3, 2, 3, 2, 3, 2, 2, 2, 7, 2, 2, 3, 3, 2, 3, 2, 2, 3, 2, 3, 2, 3, 2, 2, 2, 25, 2, 2, 3, 2, 2, 3, 3, 2, 3, 3, 2, 3, 2, 2, 3, 2

Offset: 1

Joseph Myers, Sep 04 2009

Joseph Myers, Table of n, a(n) for n=1..8989
Index entries for the Kaprekar map

Cf. A165071, A165082, A165077, A165079, A165081, A165089.

In other bases: Empty (base 2), A165005 (base 3), A165024 (base 4), A165044 (base 5), A165063 (base 6), A165102 (base 8), A165122 (base 9), A164721 (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

A165080 Consider the base-7 Kaprekar map n->K(n) defined in A165071. 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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A165003 Length of cycle mentioned in A165002.

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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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A165077 Length of cycle mentioned in A165076.

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A165079 Length of cycle mentioned in A165078.

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A165083 Length of cycle mentioned in A165082.

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A165100 Length of cycle mentioned in A165099.

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