A165075 -id:A165075 - cp's OEIS frontend

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

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).

A008722 Molien series for 3-dimensional group [2,9] = *229.

Original entry on oeis.org

1, 0, 2, 0, 3, 0, 4, 0, 5, 1, 6, 2, 7, 3, 8, 4, 9, 5, 11, 6, 13, 7, 15, 8, 17, 9, 19, 11, 21, 13, 23, 15, 25, 17, 27, 19, 30, 21, 33, 23, 36, 25, 39, 27, 42, 30, 45, 33, 48, 36, 51, 39, 54, 42, 58, 45, 62, 48, 66, 51, 70, 54, 74, 58, 78, 62, 82, 66, 86, 70, 90, 74, 95, 78, 100, 82

Offset: 0

N. J. A. Sloane

It appears that a(n) is the number of (n+11)-digit fixed points under the base-7 Kaprekar map A165071 (see A165075 for the list of fixed points). - Joseph Myers, Sep 04 2009

a(n) is the number of partitions of n into parts 2 and 9 where there are two kinds of parts 2. - Hoang Xuan Thanh, Jun 20 2025

G. C. Greubel, Table of n, a(n) for n = 0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 225
Index entries for Molien series
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1,0,0,0,0,1,0,-2,0,1).

a:=[1,0,2,0,3,0,4,0,5,1,6,2,7];; for n in [14..80] do a[n]:= 2*a[n-2] -a[n-4]+a[n-9]-2*a[n-11]+a[n-13]; od; a; # G. C. Greubel, Sep 09 2019

R:=PowerSeriesRing(Integers(), 80); Coefficients(R!( 1/((1-x^2)^2*(1-x^9)) )); // G. C. Greubel, Sep 09 2019

1/((1-x^2)^2*(1-x^9)); seq(coeff(series(%, x, n+1), x, n), n = 0..80); # modified by G. C. Greubel, Sep 09 2019

LinearRecurrence[{0,2,0,-1,0,0,0,0,1,0,-2,0,1}, {1,0,2,0,3,0,4,0,5,1,6, 2,7}, 80] (* Ray Chandler, Jul 15 2015 *)

my(x='x+O('x^80)); Vec(1/((1-x^2)^2*(1-x^9))) \\ G. C. Greubel, Sep 09 2019

def A008722_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/((1-x^2)^2*(1-x^9)) ).list()
A008722_list(80) # G. C. Greubel, Sep 09 2019

G.f.: 1/((1-x^2)^2*(1-x^9)).
a(n) = 2*a(n-2) - a(n-4) + a(n-9) - 2*a(n-11) + a(n-13). - R. J. Mathar, Dec 18 2014
a(n) = floor((n^2 + n*(13+9*(-1)^n) + 62*(-1)^n + 75)/72) - [(n mod 9)=7], where [] is Iverson bracket. - Hoang Xuan Thanh, Jun 20 2025

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).

A319798 Smallest fixed points (>0) of the base-n Kaprekar map.

Original entry on oeis.org

9, 184, 30, 8, 105, 1922263344, 21, 41520, 495, 40, 858, 3488424, 65, 30996, 2040, 96, 2907, 264925230120, 133, 2787400, 5313, 176, 6900, 237360, 225, 9742824, 10962, 280, 13485, 763713003420, 341, 26485184, 19635, 408, 23310, 107599353444576, 481, 60920080, 31980

Offset: 2

Seiichi Manyama, Sep 28 2018

Conjecture: If n = 3*k - 1 (>2), a(n) = A000567(k). For example, a(29) = 10 * (3*10 - 2) = 280.

Index entries for the Kaprekar map

Cf. A000567, A099009, A163205, A164997, A165016, A165036, A165055, A165075, A165094, A165114, A319839.

a(19) and a(31)-a(40) from Giovanni Resta, Oct 02 2018

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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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A008722 Molien series for 3-dimensional group [2,9] = *229.

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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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A319798 Smallest fixed points (>0) of the base-n Kaprekar map.

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