A163205 -id:A163205 - cp's OEIS frontend

A165049 Smallest member of cycle corresponding to n-th term of A165048.

0, 8, 48, 392, 1992, 7488, 53712, 249992, 1831056, 6249992, 45781056, 48217776, 170312312, 1144531056, 1205467776, 1217651376, 4576557032, 4514058432, 28613281056, 28671874056, 22460937432, 30136717776, 30441401376

Offset: 1

Joseph Myers, Sep 04 2009

Initial terms in base 5: 0, 13, 143, 3032, 30432, 214423, 3204322, 30444432, 432043211, 3044444432.

Joseph Myers, Table of n, a(n) for n=1..15716
Index entries for the Kaprekar map

Cf. A165032, A165037, A165048, A165050.

In other bases: A163205 (base 2), A165010 (base 3), A165030 (base 4), A165069 (base 6), A165088 (base 7), A165108 (base 8), A165128 (base 9), A151965 (base 10).

A165069 Smallest member of cycle corresponding to n-th term of A165068.

Original entry on oeis.org

0, 105, 430, 5600, 4305, 16840, 42925, 33860, 27195, 218960, 895275, 1548445, 1221860, 1657225, 1275170, 7892360, 6018495, 45962330, 55760125, 56319925, 59679145, 44002820, 60331825, 47681900, 284180120, 348285175, 277695950

Offset: 1

Joseph Myers, Sep 04 2009

Initial terms in base 6: 0, 253, 1554, 41532, 31533, 205544, 530421, 420432, 325523, 4405412.

Joseph Myers, Table of n, a(n) for n=1..23045
Index entries for the Kaprekar map

Cf. A165051, A165056, A165068, A165070.

In other bases: A163205 (base 2), A165010 (base 3), A165030 (base 4), A165049 (base 5), A165088 (base 7), A165108 (base 8), A165128 (base 9), A151965 (base 10).

A165088 Smallest member of cycle corresponding to n-th term of A165087.

Original entry on oeis.org

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

Offset: 1

Joseph Myers, Sep 04 2009

Initial terms in base 7: 0, 264, 3054, 40653, 320544, 5306532, 42166443, 530666532, 5440665222, 5431055322.

Joseph Myers, Table of n, a(n) for n=1..13070
Index entries for the Kaprekar map

Cf. A165071, A165076, A165087, A165089.

In other bases: A163205 (base 2), A165010 (base 3), A165030 (base 4), A165049 (base 5), A165069 (base 6), A165108 (base 8), A165128 (base 9), A151965 (base 10).

A165108 Smallest member of cycle corresponding to n-th term of A165107.

Original entry on oeis.org

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

Offset: 1

Joseph Myers, Sep 04 2009

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

Joseph Myers, Table of n, a(n) for n=1..8775
Index entries for the Kaprekar map

Cf. A165090, A165095, A165107, A165109.

In other bases: A163205 (base 2), A165010 (base 3), A165030 (base 4), A165049 (base 5), A165069 (base 6), A165088 (base 7), A165128 (base 9), A151965 (base 10).

A165128 Smallest member of cycle corresponding to n-th term of A165127.

Original entry on oeis.org

0, 16, 320, 2256, 3712, 34960, 41520, 183696, 3496800, 31596672, 31531872, 278474880, 326952560, 2066242576, 2516902752, 2598744000, 26531651360, 23087388720, 167365651216, 203869268832, 211282856832, 2184410558000

Offset: 1

Joseph Myers, Sep 04 2009

Initial terms in base 9: 0, 17, 385, 3076, 5074, 52854, 62853, 308876, 6518633, 65407433.

Joseph Myers, Table of n, a(n) for n=1..1560
Index entries for the Kaprekar map

Cf. A165110, A165115, A165127, A165129.

In other bases: A163205 (base 2), A165010 (base 3), A165030 (base 4), A165049 (base 5), A165069 (base 6), A165088 (base 7), A165108 (base 8), A151965 (base 10).

A164885 Length of preperiodic part of trajectory of n under iteration of the base-2 Kaprekar map in A164884.

Original entry on oeis.org

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

Offset: 0

Joseph Myers, Aug 29 2009

All base-2 cycles are fixed points, so one less than A164886.

Joseph Myers, Table of n, a(n) for n = 0..1024
Index entries for the Kaprekar map

Cf. A164884, A163205, A164886, A164887.

In other bases: A164995 (base 3), A165014 (base 4), A165034 (base 5), A165053 (base 6), A165073 (base 7), A165092 (base 8), A165112 (base 9), A151962 (base 10).

Cross-references edited by Joseph Myers, Sep 04 2009

A164886 (Length of preperiodic part) + (length of cycle) of trajectory of n under iteration of the base-2 Kaprekar map in A164884.

Original entry on oeis.org

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

Offset: 0

Joseph Myers, Aug 29 2009

All base-2 cycles are fixed points, so one more than A164885.

Joseph Myers, Table of n, a(n) for n = 0..1024
Index entries for the Kaprekar map

Cf. A164884, A163205, A164885, A164887.

In other bases: A164996 (base 3), A165015 (base 4), A165035 (base 5), A165054 (base 6), A165074 (base 7), A165093 (base 8), A165113 (base 9), A151963 (base 10).

Cross-references edited by Joseph Myers, Sep 04 2009

A164887 a(n) = smallest number that leads to a new fixed point under the base-2 Kaprekar map of A164884.

Original entry on oeis.org

0, 9, 17, 33, 35, 65, 67, 129, 131, 135, 257, 259, 263, 513, 515, 519, 527, 1025, 1027, 1031, 1039, 2049, 2051, 2055, 2063, 2079, 4097, 4099, 4103, 4111, 4127, 8193, 8195, 8199, 8207, 8223, 8255, 16385, 16387, 16391, 16399, 16415, 16447, 32769, 32771

Offset: 1

Joseph Myers, Aug 29 2009

Note that all base-2 cycles are fixed points.
Initial terms in base 2: 0,1001,10001,100001,100011,1000001,1000011,10000001,10000011,10000111.
The corresponding fixed points appear in ascending numerical order (i.e., A163205).

Joseph Myers, Table of n, a(n) for n=1..9802
Index entries for the Kaprekar map

Cf. A164884, A163205, A164885, A164886.

In other bases: A165009 (base 3), A165029 (base 4), A165048 (base 5), A165068 (base 6), A165087 (base 7), A165107 (base 8), A165127 (base 9), A151964 (base 10).

Cross-references edited by Joseph Myers, Sep 04 2009

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).

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).

cp's OEIS Frontend

A165049 Smallest member of cycle corresponding to n-th term of A165048.

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A165069 Smallest member of cycle corresponding to n-th term of A165068.

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A165088 Smallest member of cycle corresponding to n-th term of A165087.

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A165108 Smallest member of cycle corresponding to n-th term of A165107.

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A165128 Smallest member of cycle corresponding to n-th term of A165127.

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A164885 Length of preperiodic part of trajectory of n under iteration of the base-2 Kaprekar map in A164884.

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A164886 (Length of preperiodic part) + (length of cycle) of trajectory of n under iteration of the base-2 Kaprekar map in A164884.

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A164887 a(n) = smallest number that leads to a new fixed point under the base-2 Kaprekar map of A164884.

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Extensions

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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Comments

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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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