A151949 -id:A151949 - cp's OEIS frontend

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

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)

A164736 Number of n-digit cycles of length 5 under the Kaprekar map A151949.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 3, 0, 5, 0, 6, 0, 6, 1, 6, 3, 6, 5, 6, 6, 6, 6, 7, 6, 9, 6, 11, 6, 12, 6, 12, 7, 12, 9, 12, 11, 12, 12, 12, 12, 13, 12, 15, 12, 17, 12, 18, 12, 18, 13, 18, 15, 18, 17, 18, 18, 18, 18, 19, 18, 21, 18, 23, 18, 24, 18, 24, 19, 24, 21, 24, 23, 24, 24, 24

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, A164727, A164728, A164731, A164732, A164733, A164734, A164735.

Conjectures from Chai Wah Wu, Apr 13 2024: (Start)
a(n) = a(n-1) + a(n-2) - 2*a(n-3) + a(n-4) + a(n-5) - 2*a(n-6) + a(n-7) + a(n-8) - a(n-9) for n > 15.
G.f.: x^11*(x^2 + 1)*(x^2 - x + 1)/((x - 1)^2*(x + 1)*(x^6 + x^3 + 1)). (End)

A151957 Iterate the Kaprekar map of A151949 starting at the n-digit number 100...02; sequence gives the lowest number in the resulting cycle.

Original entry on oeis.org

0, 495, 6174, 62964, 420876, 7509843, 64308654, 753098643, 6431088654, 86420987532, 643330866654, 8764209875322, 64333308666654, 885432098765412, 6543331088666544, 88543320987665412, 975533110888664421

Offset: 2

Harvey P. Dale and N. J. A. Sloane, Aug 18 2009, Aug 19 2009

Joseph Myers, Table of n, a(n) for n=2..1000 [From _Joseph Myers_, Aug 21 2009]
Index entries for the Kaprekar map

See A151958 for the length of the cycles. Cf. A151949, A151955 (the trajectory of 102), A151956 (the trajectory of 1002).

A151967 Iterate the Kaprekar map of A151949 starting at the n-digit number 100...01; sequence gives the lowest number in the resulting cycle.

Original entry on oeis.org

0, 0, 6174, 62964, 420876, 7509843, 64308654, 864197532, 6431088654, 86420987532, 643310886654, 8643209876532, 43333208766666, 885432098765412, 6543331088666544, 88754320987654212, 643111110888888654

Offset: 2

N. J. A. Sloane, Aug 20 2009

			a(6) = 420876: 100001 -> 109989 -> 980811 -> 976221 -> 853542 -> 620874 -> 851742 -> 750843 -> 840852 -> 860832 -> 862632 -> 642654 -> 420876 -> 851742 -> ..., a cycle of length 7 with smallest member 420876.

Joseph Myers, Table of n, a(n) for n=2..1000 [From _Joseph Myers_, Aug 21 2009]
Index entries for the Kaprekar map

See A151968 for the length of the cycles. Cf. A151957, A151958.

Different from A151957.

(Mma program from Harvey P. Dale) To find the first 20 terms of the trajectory of 10001, for instance:
f[n_]:=Module[{idn=IntegerDigits[n],idns},idns=Sort[idn];Abs[FromDigits[ idns]-FromDigits[Reverse[idns]]]]
NestList[f,10001,20]

Extended by Joseph Myers, Aug 21 2009

A164729 Numbers belonging to cycles of length 7 under the Kaprekar map A151949.

Original entry on oeis.org

420876, 642654, 750843, 840852, 851742, 860832, 862632, 43208766, 64326654, 75308643, 84308652, 85317642, 86308632, 86326632, 4332087666, 6433266654, 7533086643, 8433086652, 8533176642, 8633086632, 8633266632, 433320876666

Offset: 1

Joseph Myers, Aug 23 2009

Joseph Myers, Table of n, a(n) for n=1..476
Index entries for the Kaprekar map

Cf. A151949, A151959, A099009, A099010, A164716, A164723, A164725, A164727, A164730.

A164730 Least element of each cycle of length 7 under the Kaprekar map A151949.

Original entry on oeis.org

420876, 43208766, 4332087666, 433320876666, 43333208766666, 4333332087666666, 433333320876666666, 43333333208766666666, 4333333332087666666666, 433333333320876666666666

Offset: 1

Joseph Myers, Aug 23 2009

Joseph Myers, Table of n, a(n) for n=1..68
Index entries for the Kaprekar map

Cf. A151949, A151959, A099009, A164718, A164720, A164724, A164726, A164728, A164729.

A151947 a(1) = 113; thereafter a(n) = (a(n-1) with digits sorted into descending order) - (a(n-1) with digits sorted into ascending order) (see the Kaprekar map, A151949).

Original entry on oeis.org

113, 198, 792, 693, 594, 495, 495, 495, 495, 495, 495, 495, 495, 495, 495, 495, 495, 495, 495, 495, 495, 495, 495, 495, 495, 495, 495, 495, 495, 495, 495, 495, 495, 495, 495, 495, 495, 495, 495, 495, 495, 495, 495, 495, 495, 495, 495, 495

Offset: 1

N. J. A. Sloane, Aug 18 2009

Converges to the fixed point 495. For the list of fixed points see A099009.

Index entries for the Kaprekar map

Cf. A099009, A151946, A151949, A151955.

A151955 a(1) = 102; thereafter a(n) = (a(n-1) with digits sorted into descending order) - (a(n-1) with digits sorted into ascending order) (see the Kaprekar map, A151949).

Original entry on oeis.org

102, 198, 792, 693, 594, 495, 495, 495, 495, 495, 495, 495, 495, 495, 495, 495, 495, 495, 495, 495, 495, 495, 495, 495, 495, 495, 495, 495, 495, 495, 495, 495, 495, 495, 495, 495, 495, 495, 495, 495, 495, 495, 495, 495, 495, 495, 495, 495

Offset: 1

Harvey P. Dale. Aug 18 2009

102 is the smallest starting value which does not converge to 0.

Converges to the fixed point 495. For the list of fixed points see A099009.

Index entries for the Kaprekar map

Cf. A099009, A151946, A151949, A151947, A151957.

A151946 a(1) = 8127; thereafter a(n) = (a(n-1) with digits sorted into descending order) - (a(n-1) with digits sorted into ascending order) (see the Kaprekar map, A151949).

Original entry on oeis.org

8127, 7443, 3996, 6264, 4176, 6174, 6174, 6174, 6174, 6174, 6174, 6174, 6174, 6174, 6174, 6174, 6174, 6174, 6174, 6174, 6174, 6174, 6174, 6174, 6174, 6174, 6174, 6174, 6174, 6174, 6174, 6174, 6174, 6174, 6174, 6174, 6174, 6174, 6174

Offset: 1

N. J. A. Sloane, Aug 18 2009

Similar in spirit to the RATS (Reverse, Add Then Sort) sequences.
Converges to the fixed point 6174. For the list of fixed points see A099009.
The initial value 1001 is the smallest number that converges to 6174 - see A151967.

Cf. A099009, A151947, A151949, A151951, A151955, A151956, A151967. A151968.

A151951 a(1) = 1113; thereafter a(n) = (a(n-1) with digits sorted into descending order) - (a(n-1) with digits sorted into ascending order) (see the Kaprekar map, A151949).

Original entry on oeis.org

1113, 1998, 8082, 8532, 6174, 6174, 6174, 6174, 6174, 6174, 6174, 6174, 6174, 6174, 6174, 6174, 6174, 6174, 6174, 6174, 6174, 6174, 6174, 6174, 6174, 6174, 6174, 6174, 6174, 6174, 6174, 6174, 6174, 6174, 6174, 6174, 6174, 6174, 6174, 6174

Offset: 1

N. J. A. Sloane, Aug 18 2009

Index entries for the Kaprekar map

Cf. A099009, A151947, A151949, A151950, A069746.

NestList[FromDigits[Reverse[Sort[IntegerDigits[#]]]]-FromDigits[Sort[ IntegerDigits[ #]]]&,1113,40] (* or *) PadRight[{1113,1998,8082,8532},40,{6174}] (* Harvey P. Dale, May 10 2021 *)

cp's OEIS Frontend

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

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A164736 Number of n-digit cycles of length 5 under the Kaprekar map A151949.

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A151957 Iterate the Kaprekar map of A151949 starting at the n-digit number 100...02; sequence gives the lowest number in the resulting cycle.

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A151967 Iterate the Kaprekar map of A151949 starting at the n-digit number 100...01; sequence gives the lowest number in the resulting cycle.

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A164729 Numbers belonging to cycles of length 7 under the Kaprekar map A151949.

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A164730 Least element of each cycle of length 7 under the Kaprekar map A151949.

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A151947 a(1) = 113; thereafter a(n) = (a(n-1) with digits sorted into descending order) - (a(n-1) with digits sorted into ascending order) (see the Kaprekar map, A151949).

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A151955 a(1) = 102; thereafter a(n) = (a(n-1) with digits sorted into descending order) - (a(n-1) with digits sorted into ascending order) (see the Kaprekar map, A151949).

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A151946 a(1) = 8127; thereafter a(n) = (a(n-1) with digits sorted into descending order) - (a(n-1) with digits sorted into ascending order) (see the Kaprekar map, A151949).

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A151951 a(1) = 1113; thereafter a(n) = (a(n-1) with digits sorted into descending order) - (a(n-1) with digits sorted into ascending order) (see the Kaprekar map, A151949).

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