A165012 -id:A165012 - 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

A165030 Smallest member of cycle corresponding to n-th term of A165029.

Original entry on oeis.org

0, 30, 201, 126, 570, 3369, 3873, 2550, 14565, 41958, 54441, 62625, 64641, 234405, 254865, 171990, 873129, 954261, 1036929, 1004193, 1044993, 3755685, 2788950, 4083345, 4165185, 15285909, 13978281, 16075425, 16399953, 16730625, 16599681

Offset: 1

Joseph Myers, Sep 04 2009

Initial terms in base 4: 0, 132, 3021, 1332, 20322, 310221, 330201, 213312, 3203211, 22033212.

Joseph Myers, Table of n, a(n) for n=1..7165
Index entries for the Kaprekar map

Cf. A165012, A165017, A165029, A165031.

In other bases: A163205 (base 2), A165010 (base 3), A165049 (base 5), A165069 (base 6), A165088 (base 7), A165108 (base 8), A165128 (base 9), A151965 (base 10).

A165031 Length of cycle corresponding to n-th term of A165029.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Joseph Myers, Sep 04 2009

Joseph Myers, Table of n, a(n) for n=1..7165
Index entries for the Kaprekar map

Cf. A165012, A165017, A165029, A165030.

In other bases: A000012 (base 2), A165011 (base 3), A165050 (base 5), A165070 (base 6), A165089 (base 7), A165109 (base 8), A165129 (base 9), A151966 (base 10).

A165020 Length of cycle mentioned in A165019.

Original entry on oeis.org

2, 2, 2, 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, 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, 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, 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..199
Index entries for the Kaprekar map

Cf. A165012, A165019, A165018, A165022, A165024, A165031.

In other bases: Empty (base 2), A165001 (base 3), A165040 (base 5), A165059 (base 6), A165079 (base 7), A165098 (base 8), A165118 (base 9), A164715 (base 10).

A165018 Length of cycle mentioned in A165017.

Original entry on oeis.org

1, 1, 2, 2, 1, 2, 2, 1, 1, 1, 1, 3, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Joseph Myers, Sep 04 2009

Joseph Myers, Table of n, a(n) for n=1..7217
Index entries for the Kaprekar map

Cf. A165012, A165017, A165020, A165022, A165024, A165031.

In other bases: A000012 (base 2), A164999 (base 3), A165038 (base 5), A165057 (base 6), A165077 (base 7), A165096 (base 8), A165116 (base 9), A164717 (base 10).

A165022 Length of cycle mentioned in A165021.

Original entry on oeis.org

1, 1, 2, 1, 2, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Joseph Myers, Sep 04 2009

Joseph Myers, Table of n, a(n) for n=1..7165
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. 16.
Index entries for the Kaprekar map

Cf. A165012, A165021, A165018, A165020, A165024, A165031.

In other bases: A000012 (base 2), A165003 (base 3), A165042 (base 5), A165061 (base 6), A165081 (base 7), A165100 (base 8), A165120 (base 9), A164719 (base 10).

A165024 Length of cycle mentioned in A165023.

Original entry on oeis.org

2, 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, 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, 3, 3, 3, 3, 3, 3, 3, 3, 3

Offset: 1

Joseph Myers, Sep 04 2009

Index entries for the Kaprekar map

Cf. A165012, A165023, A165018, A165020, A165022, A165031.

In other bases: Empty (base 2), A165005 (base 3), A165044 (base 5), A165063 (base 6), A165083 (base 7), A165102 (base 8), A165122 (base 9), A164721 (base 10).

A319652 Write n in 4-ary, sort digits into increasing order.

Original entry on oeis.org

0, 1, 2, 3, 1, 5, 6, 7, 2, 6, 10, 11, 3, 7, 11, 15, 1, 5, 6, 7, 5, 21, 22, 23, 6, 22, 26, 27, 7, 23, 27, 31, 2, 6, 10, 11, 6, 22, 26, 27, 10, 26, 42, 43, 11, 27, 43, 47, 3, 7, 11, 15, 7, 23, 27, 31, 11, 27, 43, 47, 15, 31, 47, 63, 1, 5, 6, 7, 5, 21, 22, 23, 6, 22, 26, 27

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), this sequence (b=4), A319653 (b=5), A319654 (b=6), A319655 (b=7), A319656 (b=8), A319657 (b=9), A004185 (b=10).

Cf. A165012.

a:= n-> (l-> add(l[-i]*4^(i-1), i=1..nops(l)))(sort(convert(n, base, 4))):
seq(a(n), n=0..75);  # Alois P. Heinz, Aug 07 2024

Table[FromDigits[Sort[IntegerDigits[n, 4]], 4], {n, 0, 100}] (* Paolo Xausa, Aug 07 2024 *)

a(n) = fromdigits(vecsort(digits(n, 4)), 4); \\ 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(4, 100)

A319720 Write n in 4-ary, sort digits into decreasing order.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 9, 13, 8, 9, 10, 14, 12, 13, 14, 15, 16, 20, 36, 52, 20, 21, 37, 53, 36, 37, 41, 57, 52, 53, 57, 61, 32, 36, 40, 56, 36, 37, 41, 57, 40, 41, 42, 58, 56, 57, 58, 62, 48, 52, 56, 60, 52, 53, 57, 61, 56, 57, 58, 62, 60, 61, 62, 63, 64, 80, 144, 208, 80, 84

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), this sequence (b=4), A319722 (b=5), A319723 (b=6), A319724 (b=7), A319725 (b=8), A319726 (b=9), A004186 (b=10).

Cf. A165012, A319652.

Table[FromDigits[ReverseSort[IntegerDigits[n, 4]], 4], {n, 0, 100}] (* Paolo Xausa, Aug 07 2024 *)

a(n) = fromdigits(vecsort(digits(n, 4), , 4), 4); \\ 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(4, 100)

n <= a(n) < 4n. - 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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A165030 Smallest member of cycle corresponding to n-th term of A165029.

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A165031 Length of cycle corresponding to n-th term of A165029.

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A165020 Length of cycle mentioned in A165019.

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A165018 Length of cycle mentioned in A165017.

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A165022 Length of cycle mentioned in A165021.

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A165024 Length of cycle mentioned in A165023.

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A319652 Write n in 4-ary, sort digits into increasing order.

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A319720 Write n in 4-ary, sort digits into decreasing order.

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Ruby

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