A165076 -id:A165076 - cp's OEIS frontend

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

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).

A165089 Length of cycle corresponding to n-th term of A165087.

Original entry on oeis.org

1, 2, 3, 5, 6, 6, 6, 11, 2, 2, 2, 1, 5, 2, 6, 1, 1, 2, 9, 1, 2, 1, 1, 12, 2, 2, 1, 1, 1, 1, 2, 2, 1, 4, 2, 1, 1, 1, 1, 5, 7, 1, 2, 2, 2, 3, 1, 2, 1, 2, 2, 1, 1, 1, 1, 5, 2, 1, 2, 1, 1, 2, 3, 3, 2, 1, 1, 2, 2, 1, 1, 1, 1, 1, 5, 5, 2, 1, 1, 2, 2, 1, 3, 3, 2, 2, 1, 3, 2, 1, 1, 2, 2, 2, 1, 1, 1, 1, 24, 1, 1, 2, 1, 1

Offset: 1

Joseph Myers, Sep 04 2009

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

Cf. A165071, A165076, A165087, A165088.

In other bases: A000012 (base 2), A165011 (base 3), A165031 (base 4), A165050 (base 5), A165070 (base 6), A165109 (base 8), A165129 (base 9), A151966 (base 10).

A165080 Consider the base-7 Kaprekar map n->K(n) defined in A165071. Sequence gives least elements of each cycle, including fixed points.

Original entry on oeis.org

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

Offset: 1

Joseph Myers, Sep 04 2009

Initial terms in base 7: 0, 264, 3054, 40653, 320544, 5306532, 42166443, 530666532, 5316666432, 5431055322.

Joseph Myers, Table of n, a(n) for n=1..13070
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 A165075 and A165082. Cf. A165071, A165081, A165076, A165078, A165088, A165084.

In other bases: A163205 (base 2), A165002 (base 3), A165021 (base 4), A165041 (base 5), A165060 (base 6), A165099 (base 8), A165119 (base 9), A164718 (base 10).

A165082 Consider the base-7 Kaprekar map n->K(n) defined in A165071. Sequence gives least elements of each cycle of length > 1.

Original entry on oeis.org

144, 1068, 9936, 55500, 640992, 3562968, 31412208, 220709400, 227429400, 228238488, 11150046252, 11432420652, 75796404672, 419850417612, 546394287000, 3939440152944, 26773614188652, 26884299308652

Offset: 1

Joseph Myers, Sep 04 2009

Initial terms in base 7: 264, 3054, 40653, 320544, 5306532, 42166443, 530666532, 5316666432, 5431055322, 5440665222.

Joseph Myers, Table of n, a(n) for n=1..8989
Index entries for the Kaprekar map

Cf. A165071, A165083, A165075, A165076, A165080, A165078, A165088.

In other bases: Empty (base 2), A165004 (base 3), A165023 (base 4), A165043 (base 5), A165062 (base 6), A165101 (base 8), A165121 (base 9), A164720 (base 10).

A165087 a(n) = smallest number that leads to a new cycle under the base-7 Kaprekar map of A165071.

Original entry on oeis.org

0, 51, 344, 2402, 16808, 117650, 823544, 5764802, 40353608, 40353623, 40354466, 282475250, 1977326744, 1977326763, 13841287202, 13841287211, 13841290483, 96889010408, 96889010409, 678223072850, 678223072851, 678223072995

Offset: 1

Joseph Myers, Sep 04 2009

Initial terms in base 7: 0, 102, 1001, 10001, 100001, 1000001, 10000001, 100000001, 1000000001, 1000000022.

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

Cf. A165071, A165076, A165088, A165089.

In other bases: A164887 (base 2), A165009 (base 3), A165029 (base 4), A165048 (base 5), A165068 (base 6), A165107 (base 8), A165127 (base 9), A151964 (base 10).

A165084 Number of cycles of n-digit numbers (including fixed points) under the base-7 Kaprekar map A165071.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 1, 1, 1, 3, 1, 2, 3, 2, 4, 2, 5, 1, 8, 6, 10, 5, 13, 7, 18, 7, 20, 8, 25, 12, 30, 13, 35, 15, 43, 16, 51, 20, 56, 25, 63, 29, 71, 33, 79, 38, 89, 42, 99, 48, 112, 53, 123, 57, 137, 63, 150, 71, 164, 79, 177, 87, 190, 95, 210, 101, 229, 109, 246, 119, 263, 132, 280

Offset: 1

Joseph Myers, Sep 04 2009

Joseph Myers, Table of n, a(n) for n=1..100
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. A165071, A165076, A165085, A008722.

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), A165103 (base 8), A165123 (base 9), A164731 (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).

A165086 Consider the base-7 Kaprekar map x->K(x) described in A165071. Sequence gives the smallest number that belongs to a cycle of length n under repeated iteration of this map, or -1 if there is no cycle of length n.

Original entry on oeis.org

0, 144, 1068, 9458722410775248, 9936, 55500, 65945195409025452

Offset: 1

Joseph Myers, Sep 04 2009

Known values (to 100 base-7 digits):
a(1) = 0 (base 10) = 0 (base 7)
a(2) = 144 (base 10) = 264 (base 7)
a(3) = 1068 (base 10) = 3054 (base 7)
a(4) = 9458722410775248 (base 10) = 5544222066654442212 (base 7)
a(5) = 9936 (base 10) = 40653 (base 7)
a(6) = 55500 (base 10) = 320544 (base 7)
a(7) = 65945195409025452 (base 10) = 55332221066554443312 (base 7)
a(9) = 419850417612 (base 10) = 42222166444443 (base 7)
a(10) = 114965566537586468276798389479111631100827277423731225926928273344 (base 10) = 65444444444444444444444443066666666666666666666666532222222222222222222222211 (base 7)
a(11) = 31412208 (base 10) = 530666532 (base 7)
a(12) = 26884299308652 (base 10) = 5443216666443222 (base 7)
a(13) = 894060461610805641013834968 (base 10) = 54444444322106666665544322222222 (base 7)
a(14) = 1591271424672409468790707489057394638817384701224062547077367141620193382944 (base 10) = 65444444444444444444444444444306666666666666666666666666665322222222222222222222222222211 (base 7)
a(17) = 107837050564847832079804652808012 (base 10) = 55444444332221110666655554443322222212 (base 7)
a(24) = 7598644111289477155212 (base 10) = 54443222221066554444432222 (base 7)
a(25) = 18244344524504743400068812 (base 10) = 544432222222106655444444432222 (base 7)

Index entries for the Kaprekar map

Cf. A165071, A165075, A165076, A165078, A165080, A165082.

In other bases: A153881 (base 2), A165008 (base 3), A165028 (base 4), A165047 (base 5), A165067 (base 6), A165106 (base 8), A165126 (base 9), A151959 (base 10).

cp's OEIS Frontend

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

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A165089 Length of cycle corresponding to n-th term of A165087.

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A165080 Consider the base-7 Kaprekar map n->K(n) defined in A165071. Sequence gives least elements of each cycle, including fixed points.

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A165082 Consider the base-7 Kaprekar map n->K(n) defined in A165071. Sequence gives least elements of each cycle of length > 1.

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A165087 a(n) = smallest number that leads to a new cycle under the base-7 Kaprekar map of A165071.

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A165084 Number of cycles of n-digit numbers (including fixed points) under the base-7 Kaprekar map A165071.

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

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A165086 Consider the base-7 Kaprekar map x->K(x) described in A165071. Sequence gives the smallest number that belongs to a cycle of length n under repeated iteration of this map, or -1 if there is no cycle of length n.

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