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Consider the base-8 Kaprekar map n->K(n) defined in A165090. Sequence gives numbers belonging to cycles, including fixed points.
+10
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
COMMENTS
Initial terms in base 8: 0, 25, 374, 1776, 3065, 3774, 4244, 5243, 6062, 6152.
Number of n-digit numbers in a cycle (including fixed points) under the Kaprekar map A151949.
+10
13
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
CROSSREFS
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]
Number of n-digit numbers in a cycle (including fixed points) under the base-4 Kaprekar map A165012
+10
11
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
CROSSREFS
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).
Number of n-digit numbers in a cycle (including fixed points) under the base-6 Kaprekar map A165051
+10
11
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
CROSSREFS
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).
Number of cycles of n-digit numbers (including fixed points) under the base-8 Kaprekar map A165090.
+10
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
CROSSREFS
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).
Number of n-digit numbers in a cycle (including fixed points) under the base-9 Kaprekar map A165110.
+10
11
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
CROSSREFS
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).
Number of n-digit numbers in a cycle (including fixed points) under the base-3 Kaprekar map A164993
+10
10
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
CROSSREFS
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).
Number of n-digit numbers in a cycle (including fixed points) under the base-5 Kaprekar map A165032
+10
10
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
CROSSREFS
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).
Number of n-digit numbers in a cycle (including fixed points) under the base-7 Kaprekar map A165071
+10
10
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
CROSSREFS
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).
Number of n-digit fixed points under the base-8 Kaprekar map A165090
+10
7
1, 1, 1, 0, 0, 2, 1, 1, 1, 1, 0, 4, 0, 4, 2, 2, 2, 4, 2, 3, 6, 5, 2, 7, 2, 6, 5, 10, 5, 8, 5, 8, 7, 9, 11, 12, 7, 11, 9, 12, 9, 21, 10, 14, 12, 15, 13, 18, 20, 18, 15, 20, 15, 23, 16, 30, 20, 23, 20, 26, 21, 27, 32, 29, 23, 32, 25, 32, 28, 43
FORMULA
a(n) = - a(n-1) + a(n-3) + 2*a(n-4) + 2*a(n-5) + a(n-6) - a(n-7) - 2*a(n-8) - 2*a(n-9) - a(n-10) + a(n-12) + a(n-13) for n > 15.
G.f.: x*(-x^14 + x^8 - x^5 + x^4 - x^2 - x - 1)/((x - 1)^3*(x + 1)^2*(x^2 + 1)*(x^2 + x + 1)*(x^4 + x^3 + x^2 + x + 1)). (End)
CROSSREFS
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), A165125 (base 9), A164733 (base 10).
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