A165017 -id:A165017 - cp's OEIS frontend

A165030 Smallest member of cycle corresponding to n-th term of A165029.

0, 30, 201, 126, 570, 3369, 3873, 2550, 14565, 41958, 54441, 62625, 64641, 234405, 254865, 171990, 873129, 954261, 1036929, 1004193, 1044993, 3755685, 2788950, 4083345, 4165185, 15285909, 13978281, 16075425, 16399953, 16730625, 16599681

Offset: 1

Joseph Myers, Sep 04 2009

Initial terms in base 4: 0, 132, 3021, 1332, 20322, 310221, 330201, 213312, 3203211, 22033212.

Joseph Myers, Table of n, a(n) for n=1..7165
Index entries for the Kaprekar map

Cf. A165012, A165017, A165029, A165031.

In other bases: A163205 (base 2), A165010 (base 3), A165049 (base 5), A165069 (base 6), A165088 (base 7), A165108 (base 8), A165128 (base 9), A151965 (base 10).

A165031 Length of cycle corresponding to n-th term of A165029.

Original entry on oeis.org

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

Offset: 1

Joseph Myers, Sep 04 2009

Joseph Myers, Table of n, a(n) for n=1..7165
Index entries for the Kaprekar map

Cf. A165012, A165017, A165029, A165030.

In other bases: A000012 (base 2), A165011 (base 3), A165050 (base 5), A165070 (base 6), A165089 (base 7), A165109 (base 8), A165129 (base 9), A151966 (base 10).

A165021 Consider the base-4 Kaprekar map n->K(n) defined in A165012. Sequence gives least elements of each cycle, including fixed points.

Original entry on oeis.org

0, 30, 126, 201, 570, 2550, 3369, 3873, 14565, 41958, 54441, 62625, 64641, 171990, 234405, 254865, 873129, 954261, 1004193, 1036929, 1044993, 2788950, 3755685, 4083345, 4165185, 11140950, 13978281, 15285909, 16075425, 16399953, 16599681

Offset: 1

Joseph Myers, Sep 04 2009

Initial terms in base 4: 0, 132, 1332, 3021, 20322, 213312, 310221, 330201, 3203211, 22033212.

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

Union of A165016 and A165023. Cf. A165012, A165022, A165017, A165019, A165030, A165025.

In other bases: A163205 (base 2), A165002 (base 3), A165041 (base 5), A165060 (base 6), A165080 (base 7), A165099 (base 8), A165119 (base 9), A164718 (base 10).

A165023 Consider the base-4 Kaprekar map n->K(n) defined in A165012. Sequence gives least elements of each cycle of length > 1.

Original entry on oeis.org

126, 570, 41958, 2788950, 178846038, 11451491670, 732979762518, 46912048616790, 3002392592864598, 192153469566932310, 12297827549946664278, 787061051157936166230, 50371908681484475913558

Offset: 1

Joseph Myers, Sep 04 2009

Initial terms in base 4: 1332, 20322, 22033212, 22220321112, 22222033211112, 22222203332111112, 22222220333321111112, 22222222033333211111112, 22222222203333332111111112, 22222222220333333321111111112.

Joseph Myers, Table of n, a(n) for n=1..67
Index entries for the Kaprekar map

Cf. A165012, A165024, A165016, A165017, A165021, A165019, A165030.

In other bases: Empty (base 2), A165004 (base 3), A165043 (base 5), A165062 (base 6), A165082 (base 7), A165101 (base 8), A165121 (base 9), A164720 (base 10).

A165025 Number of cycles of n-digit numbers (including fixed points) under the base-4 Kaprekar map A165012.

Original entry on oeis.org

1, 0, 1, 2, 1, 3, 1, 4, 3, 5, 4, 8, 5, 10, 8, 12, 10, 16, 12, 19, 16, 22, 19, 27, 22, 31, 27, 35, 31, 41, 35, 46, 41, 51, 46, 58, 51, 64, 58, 70, 64, 78, 70, 85, 78, 92, 85, 101, 92, 109, 101, 117, 109, 127, 117, 136, 127, 145, 136, 156, 145, 166, 156, 176, 166, 188, 176

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. A165012, A165017, A165026, A165027.

In other bases: A004526 (base 2, adjusted to start 1, 0, 0, 1, 1, ...), A165006 (base 3), A165045 (base 5), A165064 (base 6), A165084 (base 7), A165103 (base 8), A165123 (base 9), A164731 (base 10).

Conjectures from Colin Barker, Jun 01 2017: (Start)
G.f.: x*(1 - x^2 + x^3 - 2*x^6 + 3*x^8 - x^10) / ((1 - x)^3*(1 + x)^2*(1 + x + x^2)).
a(n) = 2*a(n-2) + a(n-3) - a(n-4) - 2*a(n-5) + a(n-7) for n>7.
(End)

A165026 Number of n-digit numbers in a cycle (including fixed points) under the base-4 Kaprekar map A165012.

Original entry on oeis.org

1, 0, 1, 3, 2, 3, 1, 6, 3, 5, 6, 8, 5, 12, 8, 12, 12, 16, 12, 21, 16, 22, 21, 27, 22, 33, 27, 35, 33, 41, 35, 48, 41, 51, 48, 58, 51, 66, 58, 70, 66, 78, 70, 87, 78, 92, 87, 101, 92, 111, 101, 117, 111, 127, 117, 138, 127, 145, 138, 156, 145, 168, 156, 176, 168, 188, 176

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. A165012, A165017, A165025, A165027.

In other bases: A004526 (base 2, adjusted to start 1, 0, 0, 1, 1, ...), A165007 (base 3), A165046 (base 5), A165065 (base 6), A165085 (base 7), A165104 (base 8), A165124 (base 9), A164732 (base 10).

A165029 a(n) = smallest number that leads to a new cycle under the base-4 Kaprekar map of A165012.

Original entry on oeis.org

0, 18, 65, 66, 257, 1025, 1039, 1050, 4097, 16385, 16386, 16399, 16447, 65537, 65551, 65898, 262145, 262146, 262207, 262239, 262399, 1048577, 1048578, 1048591, 1048639, 4194305, 4194310, 4194319, 4194399, 4194559, 4194687, 4195327

Offset: 1

Joseph Myers, Sep 04 2009

Initial terms in base 4: 0, 102, 1001, 1002, 10001, 100001, 100033, 100122, 1000001, 10000001.

Joseph Myers, Table of n, a(n) for n=1..7165
Index entries for the Kaprekar map

Cf. A165012, A165017, A165030, A165031.

In other bases: A164887 (base 2), A165009 (base 3), A165048 (base 5), A165068 (base 6), A165087 (base 7), A165107 (base 8), A165127 (base 9), A151964 (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).

cp's OEIS Frontend

A165030 Smallest member of cycle corresponding to n-th term of A165029.

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A165031 Length of cycle corresponding to n-th term of A165029.

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A165021 Consider the base-4 Kaprekar map n->K(n) defined in A165012. Sequence gives least elements of each cycle, including fixed points.

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A165023 Consider the base-4 Kaprekar map n->K(n) defined in A165012. Sequence gives least elements of each cycle of length > 1.

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A165025 Number of cycles of n-digit numbers (including fixed points) under the base-4 Kaprekar map A165012.

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A165026 Number of n-digit numbers in a cycle (including fixed points) under the base-4 Kaprekar map A165012.

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A165029 a(n) = smallest number that leads to a new cycle under the base-4 Kaprekar map of A165012.

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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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