A165090 -id:A165090 - cp's OEIS frontend

A164993 a(n) = image of n under the base-3 Kaprekar map n -> (n with digits sorted into descending order) - (n with digits sorted into ascending order).

0, 0, 0, 2, 0, 2, 4, 2, 0, 8, 8, 16, 8, 0, 8, 16, 8, 8, 16, 16, 16, 16, 8, 8, 16, 8, 0, 26, 32, 58, 32, 26, 52, 58, 52, 58, 32, 26, 52, 26, 0, 26, 52, 26, 32, 58, 52, 58, 52, 26, 32, 58, 32, 26, 52, 58, 64, 58, 52, 58, 64, 58, 52, 58, 52, 58, 52, 26, 32, 58, 32, 26, 64, 58, 52, 58, 32

Offset: 0

Joseph Myers, Sep 04 2009

			For n = 10, 10_10 = 101_3. So, a(10) = 110_3 - 11_3 = 12 - 4 = 8. - _Indranil Ghosh_, Feb 02 2017

Indranil Ghosh, Table of n, a(n) for n = 0..19683 (terms 0..2187 from Joseph Myers)
Index entries for the Kaprekar map

Cf. A164994.

In other bases: A164884 (base 2), A165012 (base 4), A165032 (base 5), A165051 (base 6), A165071 (base 7), A165090 (base 8), A165110 (base 9), A151949 (base 10).

a[n_] := With[{dd = IntegerDigits[n, 3]}, FromDigits[ReverseSort[dd], 3] - FromDigits[Sort[dd], 3]];
a /@ Range[0, 100] (* Jean-François Alcover, Jan 08 2020 *)

A164884 a(n) = image of n under the base-2 Kaprekar map n -> (n with digits sorted into descending order) - (n with digits sorted into ascending order).

Original entry on oeis.org

0, 0, 1, 0, 3, 3, 3, 0, 7, 9, 9, 7, 9, 7, 7, 0, 15, 21, 21, 21, 21, 21, 21, 15, 21, 21, 21, 15, 21, 15, 15, 0, 31, 45, 45, 49, 45, 49, 49, 45, 45, 49, 49, 45, 49, 45, 45, 31, 45, 49, 49, 45, 49, 45, 45, 31, 49, 45, 45, 31, 45, 31, 31, 0, 63, 93, 93, 105, 93, 105, 105, 105, 93, 105, 105

Offset: 0

Joseph Myers, Aug 29 2009

			For n = 17, 17_10 = 10001_2. So, a(17) = 11000_2 - 11_2 = 24 - 3 = 21. - _Indranil Ghosh_, Feb 01 2017

Indranil Ghosh, Table of n, a(n) for n = 0..16384 (terms 0..1024 from Joseph Myers)
Index entries for the Kaprekar map

Cf. A163205, A164885, A164886, A164887.

In other bases: A164993 (base 3), A165012 (base 4), A165032 (base 5), A165051 (base 6), A165071 (base 7), A165090 (base 8), A165110 (base 9), A151949 (base 10).

a[n_] := With[{dd = IntegerDigits[n, 2]}, FromDigits[ReverseSort[dd], 2] - FromDigits[Sort[dd], 2]];
a /@ Range[0, 100] (* Jean-François Alcover, Jan 08 2020 *)

def A164884(n):
    return int("".join(sorted(bin(n)[2:],reverse=True)),2)-int("".join(sorted(bin(n)[2:])),2) # Indranil Ghosh, Feb 01 2017

Cross-references edited by Joseph Myers, Sep 04 2009

A165108 Smallest member of cycle corresponding to n-th term of A165107.

Original entry on oeis.org

0, 21, 252, 1589, 1022, 21483, 17892, 213402, 102837, 147420, 1445787, 1707930, 1711962, 13667738, 16092433, 6589877, 110132442, 111443346, 93093147, 76545756, 878996762, 1029991697, 874802586, 421817781, 1068263553

Offset: 1

Joseph Myers, Sep 04 2009

Initial terms in base 8: 0, 25, 374, 3065, 1776, 51753, 42744, 640632, 310665, 437734.

Joseph Myers, Table of n, a(n) for n=1..8775
Index entries for the Kaprekar map

Cf. A165090, A165095, A165107, A165109.

In other bases: A163205 (base 2), A165010 (base 3), A165030 (base 4), A165049 (base 5), A165069 (base 6), A165088 (base 7), A165128 (base 9), A151965 (base 10).

A165109 Length of cycle corresponding to n-th term of A165107.

Original entry on oeis.org

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

Offset: 1

Joseph Myers, Sep 04 2009

Joseph Myers, Table of n, a(n) for n=1..8775
Index entries for the Kaprekar map

Cf. A165090, A165095, A165107, A165108.

In other bases: A000012 (base 2), A165011 (base 3), A165031 (base 4), A165050 (base 5), A165070 (base 6), A165089 (base 7), A165129 (base 9), A151966 (base 10).

A165096 Length of cycle mentioned in A165095.

Original entry on oeis.org

1, 1, 1, 5, 3, 5, 5, 3, 5, 3, 5, 2, 2, 4, 4, 4, 4, 3, 1, 3, 1, 3, 7, 7, 7, 4, 1, 7, 4, 7, 7, 7, 4, 4, 3, 3, 3, 3, 3, 3, 1, 1, 5, 5, 4, 4, 5, 5, 5, 5, 5, 4, 4, 5, 5, 5, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 6, 4, 6, 4, 2, 6, 6, 4, 2, 4, 6, 6, 3, 1, 3, 3, 3, 3, 3, 3, 4, 1, 3, 2, 3, 3, 3, 3, 1, 4, 4, 3, 3, 3, 4, 2

Offset: 1

Joseph Myers, Sep 04 2009

Joseph Myers, Table of n, a(n) for n=1..25797
Index entries for the Kaprekar map

Cf. A165090, A165095, A165098, A165100, A165102, A165109.

In other bases: A000012 (base 2), A164999 (base 3), A165018 (base 4), A165038 (base 5), A165057 (base 6), A165077 (base 7), A165116 (base 9), A164717 (base 10).

A165098 Length of cycle mentioned in A165097.

Original entry on oeis.org

5, 3, 5, 5, 3, 5, 3, 5, 2, 2, 4, 4, 4, 4, 3, 3, 3, 7, 7, 7, 4, 7, 4, 7, 7, 7, 4, 4, 3, 3, 3, 3, 3, 3, 5, 5, 4, 4, 5, 5, 5, 5, 5, 4, 4, 5, 5, 5, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 6, 4, 6, 4, 2, 6, 6, 4, 2, 4, 6, 6, 3, 3, 3, 3, 3, 3, 3, 4, 3, 2, 3, 3, 3, 3, 4, 4, 3, 3, 3, 4, 2, 3, 3, 3, 3, 3, 3, 5, 9, 9, 9, 5, 5

Offset: 1

Joseph Myers, Sep 04 2009

Joseph Myers, Table of n, a(n) for n=1..25468
Index entries for the Kaprekar map

Cf. A165090, A165097, A165096, A165100, A165102, A165109.

In other bases: Empty (base 2), A165001 (base 3), A165020 (base 4), A165040 (base 5), A165059 (base 6), A165079 (base 7), A165118 (base 9), A164715 (base 10).

A165100 Length of cycle mentioned in A165099.

Original entry on oeis.org

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

Offset: 1

Joseph Myers, Sep 04 2009

Joseph Myers, Table of n, a(n) for n=1..8775
Anthony Kay and Katrina Downes-Ward, Fixed Points and Cycles of the Kaprekar Transformation: 2. Even bases, arXiv:2408.12257 [math.CO], 2024. See p. 33.
Index entries for the Kaprekar map

Cf. A165090, A165099, A165096, A165098, A165102, A165109.

In other bases: A000012 (base 2), A165003 (base 3), A165022 (base 4), A165042 (base 5), A165061 (base 6), A165081 (base 7), A165120 (base 9), A164719 (base 10).

A165102 Length of cycle mentioned in A165101.

Original entry on oeis.org

5, 3, 2, 4, 3, 7, 4, 3, 3, 5, 5, 4, 3, 3, 3, 3, 6, 4, 2, 3, 3, 3, 3, 4, 2, 3, 3, 3, 5, 9, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 12, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2, 5, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 6, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3

Offset: 1

Joseph Myers, Sep 04 2009

Joseph Myers, Table of n, a(n) for n=1..8446
Index entries for the Kaprekar map

Cf. A165090, A165101, A165096, A165098, A165100, A165109.

In other bases: Empty (base 2), A165005 (base 3), A165024 (base 4), A165044 (base 5), A165063 (base 6), A165083 (base 7), A165122 (base 9), A164721 (base 10).

A319656 Write n in 8-ary, sort digits into increasing order.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 1, 9, 10, 11, 12, 13, 14, 15, 2, 10, 18, 19, 20, 21, 22, 23, 3, 11, 19, 27, 28, 29, 30, 31, 4, 12, 20, 28, 36, 37, 38, 39, 5, 13, 21, 29, 37, 45, 46, 47, 6, 14, 22, 30, 38, 46, 54, 55, 7, 15, 23, 31, 39, 47, 55, 63, 1, 9, 10, 11, 12, 13, 14

Offset: 0

Seiichi Manyama, Sep 25 2018

Seiichi Manyama, Table of n, a(n) for n = 0..4095

b-ary: A038573 (b=2), A038574 (b=3), A319652 (b=4), A319653 (b=5), A319654 (b=6), A319655 (b=7), this sequence (b=8), A319657 (b=9), A004185 (b=10).

Cf. A165090.

a:= n-> (l-> add(l[-i]*8^(i-1), i=1..nops(l)))(sort(convert(n, base, 8))):
seq(a(n), n=0..70);  # Alois P. Heinz, Aug 07 2024

Table[FromDigits[Sort[IntegerDigits[n, 8]], 8], {n, 0, 100}] (* Paolo Xausa, Aug 07 2024 *)

a(n) = fromdigits(vecsort(digits(n, 8)), 8); \\ Michel Marcus, Sep 25 2018

def A(k, n)
  (0..n).map{|i| i.to_s(k).split('').sort.join.to_i(k)}
end
p A(8, 100)

A319725 Write n in 8-ary, sort digits into decreasing order.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 17, 25, 33, 41, 49, 57, 16, 17, 18, 26, 34, 42, 50, 58, 24, 25, 26, 27, 35, 43, 51, 59, 32, 33, 34, 35, 36, 44, 52, 60, 40, 41, 42, 43, 44, 45, 53, 61, 48, 49, 50, 51, 52, 53, 54, 62, 56, 57, 58, 59, 60, 61, 62, 63, 64, 72, 136, 200, 264

Offset: 0

Seiichi Manyama, Sep 26 2018

Seiichi Manyama, Table of n, a(n) for n = 0..4095

b-ary: A073138 (b=2), A319651 (b=3), A319720 (b=4), A319722 (b=5), A319723 (b=6), A319724 (b=7), this sequence (b=8), A319726 (b=9), A004186 (b=10).

Cf. A165090, A319656.

Table[FromDigits[ReverseSort[IntegerDigits[n, 8]], 8], {n, 0, 100}] (* Paolo Xausa, Aug 07 2024 *)

a(n) = fromdigits(vecsort(digits(n, 8), , 4), 8); \\ Michel Marcus, Sep 26 2018

def A(k, n)
  (0..n).map{|i| i.to_s(k).split('').sort.reverse.join.to_i(k)}
end
p A(8, 100)

n <= a(n) < 8n. - Charles R Greathouse IV, Aug 07 2024

cp's OEIS Frontend

A164993 a(n) = image of n under the base-3 Kaprekar map n -> (n with digits sorted into descending order) - (n with digits sorted into ascending order).

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A164884 a(n) = image of n under the base-2 Kaprekar map n -> (n with digits sorted into descending order) - (n with digits sorted into ascending order).

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A165108 Smallest member of cycle corresponding to n-th term of A165107.

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A165109 Length of cycle corresponding to n-th term of A165107.

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A165096 Length of cycle mentioned in A165095.

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A165098 Length of cycle mentioned in A165097.

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A165100 Length of cycle mentioned in A165099.

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A165102 Length of cycle mentioned in A165101.

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A319656 Write n in 8-ary, sort digits into increasing order.

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Ruby

A319725 Write n in 8-ary, sort digits into decreasing order.

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Programs

Mathematica

PARI

Ruby

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