A165090 -id:A165090

A165095

Consider the base-8 Kaprekar map n->K(n) defined in A165090. Sequence gives numbers belonging to cycles, including fixed points.

+20
19

0, 21, 252, 1022, 1589, 2044, 2212, 2723, 3122, 3178, 3290, 17892, 20475, 21483, 21987, 25578, 26586, 102837, 147420, 177443, 213402, 217938, 1445787, 1449819, 1646442, 1707930, 1711962, 1715994, 1740690, 1752786, 1777482, 1941345

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Initial terms in base 8: 0, 25, 374, 1776, 3065, 3774, 4244, 5243, 6062, 6152.

LINKS

Joseph Myers, Table of n, a(n) for n=1..25797

Index entries for the Kaprekar map

CROSSREFS

Union of A165094 and A165097. Cf. A165090, A165096, A165099, A165101, A165108, A165104.

In other bases: A163205 (base 2), A164998 (base 3), A165017 (base 4), A165037 (base 5), A165056 (base 6), A165076 (base 7), A165115 (base 9), A164716 (base 10).

KEYWORD

base,nonn

AUTHOR

Joseph Myers, Sep 04 2009

STATUS

approved

A165094

List of fixed points of the base-8 Kaprekar map A165090

+20
17

0, 21, 252, 147420, 213402, 1711962, 16092433, 76545756, 1068263553, 39258683100, 57497839826, 58573445322, 68675650561, 3680015145170, 3749808363210, 4238480197777, 4397695901697, 20104741893852

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Initial terms in base 8: 0, 25, 374, 437734, 640632, 6417532, 75306421, 443777334, 7753064201, 444377773334.

LINKS

Joseph Myers, Table of n, a(n) for n=1..829

Index entries for the Kaprekar map

CROSSREFS

Cf. A165090, A165095, A165097, A165105.

In other bases: A163205 (base 2), A164997 (base 3), A165016 (base 4), A165036 (base 5), A165055 (base 6), A165075 (base 7), A165114 (base 9), A099009 (base 10).

KEYWORD

base,nonn

AUTHOR

Joseph Myers, Sep 04 2009

STATUS

approved

A165097

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

+20
14

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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

LINKS

Joseph Myers, Table of n, a(n) for n=1..25468

Index entries for the Kaprekar map

CROSSREFS

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).

KEYWORD

base,nonn

AUTHOR

Joseph Myers, Sep 04 2009

STATUS

approved

A165099

Consider the base-8 Kaprekar map n->K(n) defined in A165090. Sequence gives least elements of each cycle, including fixed points.

+20
13

0, 21, 252, 1022, 1589, 17892, 21483, 102837, 147420, 213402, 1445787, 1707930, 1711962, 6589877, 13667738, 16092433, 76545756, 93093147, 110132442, 111443346, 421817781, 874802586, 878996762, 1029991697, 1068263553

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Initial terms in base 8: 0, 25, 374, 1776, 3065, 42744, 51753, 310665, 437734, 640632.

LINKS

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

CROSSREFS

Union of A165094 and A165101. Cf. A165090, A165100, A165095, A165097, A165108, A165103.

In other bases: A163205 (base 2), A165002 (base 3), A165021 (base 4), A165041 (base 5), A165060 (base 6), A165080 (base 7), A165119 (base 9), A164718 (base 10).

KEYWORD

base,nonn

AUTHOR

Joseph Myers, Sep 04 2009

STATUS

approved

A165101

Consider the base-8 Kaprekar map n->K(n) defined in A165090. Sequence gives least elements of each cycle of length > 1.

+20
13

1022, 1589, 17892, 21483, 102837, 1445787, 1707930, 6589877, 13667738, 93093147, 110132442, 111443346, 421817781, 874802586, 878996762, 1029991697, 7034104602, 7168322322, 7302540042, 26996862389, 55987891610, 56256326938

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Initial terms in base 8: 1776, 3065, 42744, 51753, 310665, 5407633, 6407632, 31106665, 64106632, 543076433.

LINKS

Joseph Myers, Table of n, a(n) for n=1..8446

Index entries for the Kaprekar map

CROSSREFS

Cf. A165090, A165102, A165094, A165095, A165099, A165097, A165108.

In other bases: Empty (base 2), A165004 (base 3), A165023 (base 4), A165043 (base 5), A165062 (base 6), A165082 (base 7), A165121 (base 9), A164720 (base 10).

KEYWORD

base,nonn

AUTHOR

Joseph Myers, Sep 04 2009

STATUS

approved

A165103

Number of cycles of n-digit numbers (including fixed points) under the base-8 Kaprekar map A165090.

+20
11

1, 1, 1, 2, 2, 3, 3, 3, 4, 5, 3, 13, 2, 16, 3, 21, 5, 34, 5, 45, 12, 62, 13, 82, 22, 104, 35, 137, 45, 170, 61, 215, 82, 264, 111, 323, 139, 389, 179, 466, 223, 564, 275, 657, 338, 774, 410, 905, 498, 1048, 587, 1212, 696, 1392, 818, 1598, 958, 1811, 1110, 2058, 1281

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,4

LINKS

Joseph Myers, Table of n, a(n) for n=1..70

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

CROSSREFS

Cf. A165090, A165095, A165104, A165105.

In other bases: A004526 (base 2, adjusted to start 1, 0, 0, 1, 1, ...), A165006 (base 3), A165025 (base 4), A165045 (base 5), A165064 (base 6), A165084 (base 7), A165123 (base 9), A164731 (base 10).

KEYWORD

base,nonn

AUTHOR

Joseph Myers, Sep 04 2009

STATUS

approved

A165104

Number of n-digit numbers in a cycle (including fixed points) under the base-8 Kaprekar map A165090

+20
11

1, 1, 1, 8, 6, 5, 12, 7, 15, 13, 12, 31, 14, 40, 14, 59, 12, 96, 11, 137, 24, 184, 35, 234, 64, 300, 102, 391, 134, 494, 175, 630, 232, 782, 311, 954, 402, 1147, 521, 1374, 659, 1649, 814, 1943, 992, 2294, 1203, 2687, 1454, 3117, 1731, 3597, 2060, 4130, 2430, 4734

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,4

LINKS

Joseph Myers, Table of n, a(n) for n=1..70

Index entries for the Kaprekar map

CROSSREFS

Cf. A165090, A165095, A165103, A165105.

In other bases: A004526 (base 2, adjusted to start 1, 0, 0, 1, 1, ...), A165007 (base 3), A165026 (base 4), A165046 (base 5), A165065 (base 6), A165085 (base 7), A165124 (base 9), A164732 (base 10).

KEYWORD

base,nonn

AUTHOR

Joseph Myers, Sep 04 2009

STATUS

approved

A165107

a(n) = smallest number that leads to a new cycle under the base-8 Kaprekar map of A165090

+20
11

0, 12, 66, 513, 514, 4097, 4124, 32769, 32779, 32868, 262145, 262148, 262291, 2097153, 2097155, 2097227, 16777217, 16777218, 16777219, 16782116, 134217729, 134217731, 134217739, 134218315, 134219007, 1073741825, 1073741826

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Initial terms in base 8: 0, 14, 102, 1001, 1002, 10001, 10034, 100001, 100013, 100144.

LINKS

Joseph Myers, Table of n, a(n) for n=1..8775

Index entries for the Kaprekar map

CROSSREFS

Cf. A165090, A165095, A165108, A165109.

In other bases: A164887 (base 2), A165009 (base 3), A165029 (base 4), A165048 (base 5), A165068 (base 6), A165087 (base 7), A165127 (base 9), A151964 (base 10).

KEYWORD

base,nonn

AUTHOR

Joseph Myers, Sep 04 2009

STATUS

approved

A165092

Length of preperiodic part of trajectory of n under iteration of the base-8 Kaprekar map in A165090

+20
10

0, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 4, 1, 3, 3, 2, 4, 2, 1, 2, 4, 0, 3, 3, 1, 4, 2, 1, 2, 4, 1, 3, 3, 1, 4, 2, 1, 2, 4, 1, 3, 3, 1, 4, 2, 1, 2, 4, 2, 3, 3, 1, 4, 2, 1, 2, 4, 2, 3, 3, 1, 4, 2, 1, 2, 2, 4, 3, 1, 2, 3, 4, 2, 1, 2, 4, 3, 1, 2, 3, 4, 2, 2, 4, 3, 1, 2, 3, 3, 4, 4, 4, 3, 1, 2, 3, 1, 3, 3, 3, 3, 1, 2, 3, 2

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,9

LINKS

Joseph Myers, Table of n, a(n) for n=0..4096

Index entries for the Kaprekar map

CROSSREFS

Cf. A165090, A165093.

In other bases: A164885 (base 2), A164995 (base 3), A165014 (base 4), A165034 (base 5), A165053 (base 6), A165073 (base 7), A165112 (base 9), A151962 (base 10).

KEYWORD

base,nonn

AUTHOR

Joseph Myers, Sep 04 2009

STATUS

approved

A165093

(Length of preperiodic part) + (length of cycle) of trajectory of n under iteration of the base-8 Kaprekar map in A165090

+20
10

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

Joseph Myers, Table of n, a(n) for n=0..4096

Index entries for the Kaprekar map

CROSSREFS

Cf. A165090, A165092.

In other bases: A164886 (base 2), A164996 (base 3), A165015 (base 4), A165035 (base 5), A165054 (base 6), A165074 (base 7), A165113 (base 9), A151963 (base 10).

KEYWORD

base,nonn

AUTHOR

Joseph Myers, Sep 04 2009

STATUS

approved