A164736 -id:A164736 - 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]

A164732 Number of n-digit numbers in a cycle (including fixed points) under the Kaprekar map A151949.

Original entry on oeis.org

1, 0, 1, 1, 10, 9, 8, 12, 16, 22, 14, 42, 18, 73, 29, 125, 34, 199, 38, 308, 49, 462, 71, 665, 105, 920, 161, 1243, 249, 1658, 379, 2170, 555, 2806, 780, 3587, 1075, 4539, 1449, 5689, 1922, 7059, 2516, 8677, 3252, 10566, 4156, 12774, 5255, 15337, 6578, 18300

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, A164731, A164733, A164734, A164735, A164736.

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), A165124 (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)

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)

A164735 Number of n-digit cycles of length 3 under the Kaprekar map A151949.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 4, 0, 10, 0, 20, 0, 36, 0, 60, 1, 94, 4, 141, 10, 204, 21, 286, 39, 392, 66, 527, 105, 696, 159, 906, 231, 1164, 326, 1477, 449, 1854, 605, 2304, 801, 2836, 1044, 3462, 1341, 4194, 1701, 5044, 2133, 6027, 2646, 7158, 3252, 8452, 3963

Offset: 1

Joseph Myers, Aug 23 2009

Joseph Myers, Table of n, a(n) for n = 1..70
M. Kauers and C. Koutschan, Some D-finite and some possibly D-finite sequences in the OEIS, arXiv:2303.02793 [cs.SC], 2023. [see page 45]
M. Kauers and C. Koutschan, Conjectured closed form for a(n), a quasi-polynomial of period 18 and degree 5.
Index entries for the Kaprekar map

Cf. A151949, A164725, A164726, A164731, A164732, A164733, A164734, A164736.

Conjectures from Chai Wah Wu, Apr 13 2024: (Start)
a(n) = 4*a(n-2) - 6*a(n-4) + 5*a(n-6) - 5*a(n-8) + a(n-9) + 6*a(n-10) - 4*a(n-11) - 4*a(n-12) + 6*a(n-13) + a(n-14) - 5*a(n-15) + 5*a(n-17) - 6*a(n-19) + 4*a(n-21) - a(n-23) for n > 25.
G.f.: x*(-x^24 + x^22 + x^18 - x^16 + x^15 - x^13 + x^7)/((x - 1)^6*(x + 1)^5*(x^2 - x + 1)*(x^2 + x + 1)^2*(x^6 + x^3 + 1)). (End)

cp's OEIS Frontend

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

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A164732 Number of n-digit numbers in a cycle (including fixed points) under the Kaprekar map A151949.

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

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

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A164735 Number of n-digit cycles of length 3 under the Kaprekar map A151949.

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