A164732 -id:A164732 - cp's OEIS frontend

A164731 Number of cycles of n-digit numbers (including fixed points) under the Kaprekar map A151949.

1, 0, 1, 1, 3, 3, 1, 4, 3, 8, 3, 16, 5, 27, 8, 46, 9, 73, 11, 110, 16, 162, 25, 231, 37, 318, 58, 429, 88, 572, 132, 747, 192, 963, 269, 1229, 372, 1551, 500, 1939, 662, 2401, 864, 2948, 1115, 3586, 1421, 4330, 1792, 5194, 2240, 6191, 2764, 7338, 3382, 8650, 4105

Offset: 1

Joseph Myers, Aug 23 2009

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

Cf. A151949, A164716, A164732, A164733, A164734, A164735, A164736.

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), A165103 (base 8), A165123 (base 9). [Joseph Myers, Sep 05 2009]

A164733 Number of n-digit fixed points under the Kaprekar map A151949.

Original entry on oeis.org

1, 0, 1, 1, 0, 2, 0, 2, 2, 3, 1, 5, 1, 6, 2, 8, 2, 12, 3, 14, 5, 17, 7, 21, 8, 25, 12, 30, 14, 36, 17, 43, 21, 49, 25, 58, 31, 66, 36, 75, 43, 85, 49, 96, 58, 109, 66, 121, 75, 136, 86, 150, 96, 167, 109, 184, 121, 202, 136, 222, 150, 242, 167, 265, 185, 287, 202, 313, 222, 338

Offset: 1

Joseph Myers, Aug 23 2009

Joseph Myers, Table of n, a(n) for n=1..70
Index entries for the Kaprekar map

Cf. A151949, A099009, A164731, A164732, A164734, A164735, A164736.
Bisections: A309223, A309224.
In other bases: A004526 (base 2, adjusted to start 1, 0, 0, 1, 1, ...), A008615 (base 3), A165027 (base 4), A008617 (base 5), A165066 (base 6), A008722 (base 7, conjecturally), A165105 (base 8), A165125 (base 9). [From Joseph Myers, Sep 05 2009]

Conjectures from Chai Wah Wu, Apr 13 2024: (Start)
a(n) = a(n-2) + a(n-6) - a(n-8) + a(n-9) - a(n-11) + a(n-14) - a(n-15) - a(n-16) + a(n-17) - a(n-20) + a(n-22) - a(n-23) + a(n-25) + a(n-29) - a(n-31) for n > 33.
G.f.: x*(-x^32 + x^31 - x^29 + x^28 - x^27 + x^26 - x^24 + 2*x^23 - x^22 + x^21 + x^20 + 2*x^18 - x^17 + x^16 + 2*x^15 - 3*x^14 + 2*x^13 - x^12 + x^11 - x^9 + 2*x^8 - x^6 + x^5 - x^4 + x^3 + 1)/(x^31 - x^29 - x^25 + x^23 - x^22 + x^20 - x^17 + x^16 + x^15 - x^14 + x^11 - x^9 + x^8 - x^6 - x^2 + 1). (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).

A165065 Number of n-digit numbers in a cycle (including fixed points) under the base-6 Kaprekar map A165051.

Original entry on oeis.org

1, 0, 1, 6, 3, 6, 2, 13, 3, 10, 4, 15, 6, 21, 10, 29, 13, 40, 18, 52, 26, 65, 35, 82, 46, 101, 59, 123, 74, 149, 92, 176, 114, 207, 138, 243, 164, 282, 195, 325, 229, 372, 267, 423, 310, 479, 355, 541, 405, 607, 461, 677, 521, 754, 586, 836, 656, 924, 731, 1019, 812

Offset: 1

Joseph Myers, Sep 04 2009

Joseph Myers, Table of n, a(n) for n=1..100
Index entries for the Kaprekar map

Cf. A165051, A165056, A165064, A165066.

In other bases: A004526 (base 2, adjusted to start 1, 0, 0, 1, 1, ...), A165007 (base 3), A165026 (base 4), A165046 (base 5), A165085 (base 7), A165104 (base 8), A165124 (base 9), A164732 (base 10).

A165104 Number of n-digit numbers in a cycle (including fixed points) under the base-8 Kaprekar map A165090.

Original entry on oeis.org

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

Offset: 1

Joseph Myers, Sep 04 2009

Joseph Myers, Table of n, a(n) for n=1..70
Index entries for the Kaprekar map

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).

A165124 Number of n-digit numbers in a cycle (including fixed points) under the base-9 Kaprekar map A165110.

Original entry on oeis.org

1, 2, 2, 6, 6, 14, 2, 5, 13, 10, 7, 10, 11, 13, 14, 3, 11, 17, 11, 23, 23, 17, 28, 10, 58, 17, 69, 40, 71, 38, 83, 43, 125, 26, 139, 17, 154, 23, 167, 42, 202, 71, 204, 34, 209, 65, 227, 45, 212, 62, 223, 86, 243, 77, 317, 90, 310, 125, 324, 83, 363, 86, 346, 159, 492, 124

Offset: 1

Joseph Myers, Sep 04 2009

Joseph Myers, Table of n, a(n) for n=1..70
Index entries for the Kaprekar map

Cf. A165110, A165115, A165123, A165125.

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), A165104 (base 8), A164732 (base 10).

A165007 Number of n-digit numbers in a cycle (including fixed points) under the base-3 Kaprekar map A164993.

Original entry on oeis.org

1, 0, 0, 2, 1, 3, 3, 1, 6, 5, 10, 3, 15, 4, 21, 7, 28, 12, 36, 11, 45, 13, 55, 18, 66, 19, 78, 25, 91, 28, 105, 35, 120, 41, 136, 43, 153, 47, 171, 56, 190, 58, 210, 68, 231, 73, 253, 84, 276, 85, 300, 97, 325, 103, 351, 116, 378, 119, 406, 133, 435, 140, 465, 155, 496, 162

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. A164993, A164998, A165006, A008615.

In other bases: A004526 (base 2, adjusted to start 1, 0, 0, 1, 1, ...), A165026 (base 4), A165046 (base 5), A165065 (base 6), A165085 (base 7), A165104 (base 8), A165124 (base 9), A164732 (base 10).

A165046 Number of n-digit numbers in a cycle (including fixed points) under the base-5 Kaprekar map A165032.

Original entry on oeis.org

1, 1, 2, 1, 4, 5, 4, 6, 1, 4, 4, 8, 9, 9, 21, 10, 29, 6, 48, 4, 66, 14, 93, 20, 126, 19, 166, 17, 214, 27, 270, 34, 339, 36, 410, 46, 495, 55, 591, 70, 699, 82, 819, 97, 952, 107, 1103, 135, 1260, 142, 1436, 175, 1628, 183, 1836, 231, 2061, 232, 2304, 288, 2565, 300

Offset: 1

Joseph Myers, Sep 04 2009

Joseph Myers, Table of n, a(n) for n=1..100
Index entries for the Kaprekar map

Cf. A165032, A165037, A165045, A008617.

In other bases: A004526 (base 2, adjusted to start 1, 0, 0, 1, 1, ...), A165007 (base 3), A165026 (base 4), A165065 (base 6), A165085 (base 7), A165104 (base 8), A165124 (base 9), A164732 (base 10).

A165085 Number of n-digit numbers in a cycle (including fixed points) under the base-7 Kaprekar map A165071.

Original entry on oeis.org

1, 0, 2, 3, 5, 6, 6, 6, 11, 6, 1, 7, 8, 11, 5, 14, 6, 2, 13, 19, 15, 11, 21, 17, 31, 32, 35, 16, 45, 42, 55, 31, 65, 34, 84, 24, 107, 45, 117, 39, 138, 54, 164, 53, 184, 67, 212, 70, 240, 81, 277, 90, 310, 97, 350, 108, 388, 125, 436, 140, 481, 155, 526, 179, 591, 181, 653

Offset: 1

Joseph Myers, Sep 04 2009

Joseph Myers, Table of n, a(n) for n=1..100
Index entries for the Kaprekar map

Cf. A165071, A165076, A165084, A008722.

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), A165104 (base 8), A165124 (base 9), A164732 (base 10).

A164734 Number of n-digit cycles of length 2 under the Kaprekar map A151949.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 2, 0, 2, 0, 1, 0, 1, 2, 1, 1, 1, 1, 1, 0, 3, 0, 3, 0, 2, 0, 2, 3, 1, 2, 1, 2, 1, 1, 4, 1, 4, 0, 3, 0, 3, 4, 2, 3, 2, 3, 1, 2, 5, 2, 5, 1, 4, 1, 4, 5, 3, 4, 3, 4, 2, 3, 7, 3, 6, 2, 5, 2, 5, 7, 4, 6, 4, 5, 3, 4, 9, 4, 8, 3, 7, 3, 6, 9, 5, 8, 5, 7, 4

Offset: 1

Joseph Myers, Aug 23 2009

Joseph Myers, Table of n, a(n) for n=1..140
Index entries for the Kaprekar map

Cf. A151949, A164723, A164724, A164731, A164732, A164733, A164735, A164736.

Conjectures from Chai Wah Wu, Apr 13 2024: (Start)
a(n) = a(n-7) + a(n-9) + a(n-14) - a(n-16) - a(n-21) - a(n-23) + a(n-30) for n > 41.
G.f.: x*(-x^40 - x^38 - x^36 + x^33 - x^32 + 2*x^31 - x^30 + 2*x^29 - x^28 + x^27 + x^25 - x^24 + 2*x^23 - x^22 + 2*x^21 - 2*x^20 + x^19 - x^16 - x^15 - x^14 + x^13 - x^12 + x^11 - x^4)/(x^30 - x^23 - x^21 - x^16 + x^14 + x^9 + x^7 - 1). (End)

cp's OEIS Frontend

A164731 Number of cycles of n-digit numbers (including fixed points) under the Kaprekar map A151949.

Views

Author

Keywords

Links

Crossrefs

A164733 Number of n-digit fixed points under the Kaprekar map A151949.

Views

Author

Keywords

Links

Crossrefs

Formula

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

Views

Author

Keywords

Links

Crossrefs

A165065 Number of n-digit numbers in a cycle (including fixed points) under the base-6 Kaprekar map A165051.

Views

Author

Keywords

Links

Crossrefs

A165104 Number of n-digit numbers in a cycle (including fixed points) under the base-8 Kaprekar map A165090.

Views

Author

Keywords

Links

Crossrefs

A165124 Number of n-digit numbers in a cycle (including fixed points) under the base-9 Kaprekar map A165110.

Views

Author

Keywords

Links

Crossrefs

A165007 Number of n-digit numbers in a cycle (including fixed points) under the base-3 Kaprekar map A164993.

Views

Author

Keywords

Links

Crossrefs

A165046 Number of n-digit numbers in a cycle (including fixed points) under the base-5 Kaprekar map A165032.

Views

Author

Keywords

Links

Crossrefs

A165085 Number of n-digit numbers in a cycle (including fixed points) under the base-7 Kaprekar map A165071.

Views

Author

Keywords

Links

Crossrefs

A164734 Number of n-digit cycles of length 2 under the Kaprekar map A151949.

Views

Author

Keywords

Links

Crossrefs

Formula