A164886 -id:A164886 - cp's OEIS frontend

A163205 The non-repetitive Kaprekar binary numbers in decimal.

0, 9, 21, 45, 49, 93, 105, 189, 217, 225, 381, 441, 465, 765, 889, 945, 961, 1533, 1785, 1905, 1953, 3069, 3577, 3825, 3937, 3969, 6141, 7161, 7665, 7905, 8001, 12285, 14329, 15345, 15841, 16065, 16129, 24573, 28665, 30705, 31713, 32193, 32385

Offset: 1

Damir Olejar, Jul 23 2009

Same as A160761, but with no repetitions. The numbers also exist in A143088, except that every first and last number is omitted from A143088's pyramid.
From Joseph Myers, Aug 29 2009: (Start)
Note that all base-2 cycles are fixed points.
Initial terms in base 2: 0, 1001, 10101, 101101, 110001, 1011101, 1101001, 10111101, 11011001, 11100001. (End)

			The number 9 is 1001 in binary. The maximum number using the same number of 0's and 1's is found and the minimum number having the same number of 0's and 1's is found to obtain the equation such as 1100 - 0011 = 1001. Repeating the same procedure gives us the same number and pattern of 0's and 1's. Therefore 9 is one of the Kaprekar numbers. If 9 did not occur before, it is counted as a number that belongs to a sequence and added to a database to skip repetitions. Numbers that end the procedure in 0 are excluded since they are not Kaprekar numbers. A number 9 can also be obtained with, let's say, 1100. Since number 9 already occurred for 1001, the number 9 occurring for 1100 is ignored to avoid repetition.

M. Charosh, Some Applications of Casting Out 999...'s, Journal of Recreational Mathematics 14, 1981-82, pp. 111-118.
D. R. Kaprekar, On Kaprekar numbers, J. Rec. Math., 13 (1980-1981), pp. 81-82.

Joseph Myers, Table of n, a(n) for n = 1..9802 [From _Joseph Myers_, Aug 29 2009]
Juergen Koeller, The Kaprekar Number
Wikipedia, Kaprekar Number
Index entries for the Kaprekar map

Cf. A160761, A143088, A164884, A164885, A164886, A164887.

In other bases: A164997 (base 3), A165016 (base 4), A165036 (base 5), A165055 (base 6), A165075 (base 7), A165094 (base 8), A165114 (base 9), A099009 (base 10).

import java.util.*; class pattern { public static void main(String args[]) { int mem1 = 0; int mem2 =1; ArrayList memory = new ArrayList(); for (int i = 1; i

nmax = 10^5; f[n_] := Module[{id, sid, min, max}, id = IntegerDigits[n, 2]; min = FromDigits[sid = Sort[id], 2]; max = FromDigits[Reverse[sid], 2]; max - min]; Reap[Do[If[(fpn = FixedPoint[f, n]) > 0, Sow[fpn]], {n, 0, nmax}]][[2, 1]] // Union // Prepend[#, 0]& (* Jean-François Alcover, Apr 23 2017 *)

Sort all integers from the number in descending order.
Sort all integers from the number in ascending order.
Subtract ascending from descending order to obtain a new number.
Repeat the steps 1-3 with a new number until a repetitive sequence is obtained or until a zero is obtained.
Call the repetitive sequence's number a Kaprekar number, ignore zeros and repetitions from the set of the final results.

Initial zero added for consistency with other bases by Joseph Myers, Aug 29 2009

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

A164885 Length of preperiodic part of trajectory of n under iteration of the base-2 Kaprekar map in A164884.

Original entry on oeis.org

0, 1, 2, 1, 2, 2, 2, 1, 2, 0, 1, 2, 1, 2, 2, 1, 2, 1, 1, 1, 1, 0, 1, 2, 1, 1, 1, 2, 1, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 1, 0, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 2, 1, 2, 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, 0, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 0

Joseph Myers, Aug 29 2009

All base-2 cycles are fixed points, so one less than A164886.

Joseph Myers, Table of n, a(n) for n = 0..1024
Index entries for the Kaprekar map

Cf. A164884, A163205, A164886, A164887.

In other bases: A164995 (base 3), A165014 (base 4), A165034 (base 5), A165053 (base 6), A165073 (base 7), A165092 (base 8), A165112 (base 9), A151962 (base 10).

Cross-references edited by Joseph Myers, Sep 04 2009

A164887 a(n) = smallest number that leads to a new fixed point under the base-2 Kaprekar map of A164884.

Original entry on oeis.org

0, 9, 17, 33, 35, 65, 67, 129, 131, 135, 257, 259, 263, 513, 515, 519, 527, 1025, 1027, 1031, 1039, 2049, 2051, 2055, 2063, 2079, 4097, 4099, 4103, 4111, 4127, 8193, 8195, 8199, 8207, 8223, 8255, 16385, 16387, 16391, 16399, 16415, 16447, 32769, 32771

Offset: 1

Joseph Myers, Aug 29 2009

Note that all base-2 cycles are fixed points.
Initial terms in base 2: 0,1001,10001,100001,100011,1000001,1000011,10000001,10000011,10000111.
The corresponding fixed points appear in ascending numerical order (i.e., A163205).

Joseph Myers, Table of n, a(n) for n=1..9802
Index entries for the Kaprekar map

Cf. A164884, A163205, A164885, A164886.

In other bases: A165009 (base 3), A165029 (base 4), A165048 (base 5), A165068 (base 6), A165087 (base 7), A165107 (base 8), A165127 (base 9), A151964 (base 10).

Cross-references edited by Joseph Myers, Sep 04 2009

A151963 (Length of preperiodic part) + (length of cycle) of trajectory of n under iteration of the Kaprekar map in A151949.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Aug 19 2009

Equals A151962(n) + 1 iff n < 10001 (when a cycle of length greater than 1 occurs for the first time).

			13->18->63->27->45->9->0->0, so a(13)=6+1 = 7.

Joseph Myers and Robert G. Wilson v, Table of n, a(n) for n = 0..1000 .
Index entries for the Kaprekar map

In other bases: A164886 (base 2), A164996 (base 3), A165015 (base 4), A165035 (base 5), A165054 (base 6), A165074 (base 7), A165093 (base 8), A165113 (base 9). - Joseph Myers, Sep 05 2009

# Maple program from R. J. Mathar:
A151949 := proc(n)
local tup;
tup := sort(convert(n,base,10)) ;
add( (op(i,tup)-op(-i,tup)) *10^(i-1),i=1..nops(tup)) :
end:
A151963 := proc(n)
local tra,x ;
tra := [n] ;
x := n ;
while true do
x := A151949(x) ;
if x in tra then
RETURN(nops(tra)) ;
fi;
tra := [op(tra),x] :
od:
end:
seq(A151963(n),n=0..120) ;

f[n_] := Module[{idn = IntegerDigits@n, idns}, idns = Sort@ idn; FromDigits@ Reverse@ idns - FromDigits@ idns]; g[n_] := Length[ NestWhileList[ f, n, UnsameQ, All]] - 1; Table[g@n, {n, 0, 104}] (* Robert G. Wilson v, Aug 20 2009 *)

Typos corrected by Joseph Myers, Aug 20 2009

More terms from R. J. Mathar and Robert G. Wilson v, Aug 20 2009

A164996 (Length of preperiodic part) + (length of cycle) of trajectory of n under iteration of the base-3 Kaprekar map in A164993.

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 3, 3, 2, 3, 3, 4, 3, 2, 3, 4, 3, 3, 4, 4, 4, 4, 3, 3, 4, 3, 2, 3, 3, 4, 3, 3, 2, 4, 3, 4, 3, 3, 3, 3, 2, 3, 3, 3, 3, 4, 3, 4, 3, 3, 3, 4, 2, 3, 3, 4, 4, 4, 3, 4, 4, 4, 3, 4, 3, 4, 3, 3, 3, 4, 3, 3, 4, 4, 3, 4, 3, 3, 3, 3, 2, 3, 3, 2, 3, 3, 2, 2, 2, 3, 3, 3, 2, 3, 3, 4, 2, 4, 2, 2, 2, 3, 2, 4, 2

Offset: 0

Joseph Myers, Sep 04 2009

Joseph Myers, Table of n, a(n) for n=0..2187
Index entries for the Kaprekar map

Cf. A164993, A164995.

In other bases: A164886 (base 2), A165015 (base 4), A165035 (base 5), A165054 (base 6), A165074 (base 7), A165093 (base 8), A165113 (base 9), A151963 (base 10).

A165015 (Length of preperiodic part) + (length of cycle) of trajectory of n under iteration of the base-4 Kaprekar map in A165012.

Original entry on oeis.org

1, 2, 2, 2, 3, 2, 3, 4, 4, 3, 2, 3, 4, 4, 3, 2, 3, 3, 2, 3, 3, 2, 3, 2, 2, 3, 3, 2, 3, 2, 1, 2, 2, 2, 2, 3, 2, 3, 3, 2, 2, 3, 2, 3, 3, 2, 3, 3, 3, 3, 3, 3, 3, 2, 2, 2, 3, 2, 3, 3, 3, 2, 3, 2, 3, 3, 3, 2, 3, 3, 3, 4, 3, 3, 3, 2, 2, 4, 2, 4, 3, 3, 3, 4, 3, 2, 3, 3, 3, 3, 3, 3, 4, 3, 3, 4, 3, 3, 3, 2, 3, 3, 3, 3, 3

Offset: 0

Joseph Myers, Sep 04 2009

Joseph Myers, Table of n, a(n) for n = 0..1024
Index entries for the Kaprekar map

Cf. A165012, A165014.

In other bases: A164886 (base 2), A164996 (base 3), A165035 (base 5), A165054 (base 6), A165074 (base 7), A165093 (base 8), A165113 (base 9), A151963 (base 10).

A165035 (Length of preperiodic part) + (length of cycle) of trajectory of n under iteration of the base-5 Kaprekar map in A165032.

Original entry on oeis.org

1, 2, 2, 2, 2, 3, 2, 3, 1, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 3, 3, 2, 3, 2, 3, 3, 3, 3, 4, 3, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 3, 3, 2, 3, 3, 3, 3, 3, 4, 3, 3, 3, 3, 3, 3, 3, 2, 3, 3, 3, 3, 3, 3, 3, 4, 3, 2, 3, 3, 3, 3, 3, 3, 4, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2, 3, 4, 3, 3, 3, 3, 4, 4, 4, 4, 4

Offset: 0

Joseph Myers, Sep 04 2009

Joseph Myers, Table of n, a(n) for n=0..3125
Index entries for the Kaprekar map

Cf. A165032, A165034.

In other bases: A164886 (base 2), A164996 (base 3), A165015 (base 4), A165054 (base 6), A165074 (base 7), A165093 (base 8), A165113 (base 9), A151963 (base 10).

A165054 (Length of preperiodic part) + (length of cycle) of trajectory of n under iteration of the base-6 Kaprekar map in A165051.

Original entry on oeis.org

1, 2, 2, 2, 2, 2, 3, 2, 3, 5, 4, 4, 5, 3, 2, 3, 5, 4, 4, 5, 3, 2, 3, 5, 4, 4, 5, 3, 2, 3, 5, 4, 4, 5, 3, 2, 3, 3, 4, 2, 3, 4, 3, 2, 3, 4, 2, 3, 4, 3, 3, 4, 2, 3, 2, 4, 4, 4, 2, 3, 3, 2, 2, 2, 2, 3, 4, 3, 3, 3, 3, 3, 4, 4, 4, 2, 3, 4, 4, 3, 3, 4, 2, 3, 4, 3, 2, 3, 4, 2, 2, 4, 3, 3, 4, 2, 3, 2, 4, 4, 4, 2, 4, 3, 2

Offset: 0

Joseph Myers, Sep 04 2009

Joseph Myers, Table of n, a(n) for n=0..1296
Index entries for the Kaprekar map

Cf. A165051, A165053.

In other bases: A164886 (base 2), A164996 (base 3), A165015 (base 4), A165035 (base 5), A165074 (base 7), A165093 (base 8), A165113 (base 9), A151963 (base 10).

A165074 (Length of preperiodic part) + (length of cycle) of trajectory of n under iteration of the base-7 Kaprekar map in A165071.

Original entry on oeis.org

1, 2, 2, 2, 2, 2, 2, 3, 2, 3, 4, 5, 3, 5, 4, 3, 2, 3, 4, 5, 3, 5, 4, 3, 2, 3, 4, 5, 3, 5, 4, 3, 2, 3, 4, 5, 3, 5, 4, 3, 2, 3, 4, 5, 3, 5, 4, 3, 2, 3, 3, 5, 3, 3, 4, 5, 3, 2, 3, 5, 3, 3, 4, 5, 3, 3, 5, 3, 3, 4, 3, 5, 5, 5, 3, 3, 4, 3, 3, 3, 3, 3, 3, 4, 4, 3, 3, 3, 3, 3, 4, 5, 4, 4, 4, 4, 4, 4, 5, 5, 5, 3, 3, 4, 5

Offset: 0

Joseph Myers, Sep 04 2009

Joseph Myers, Table of n, a(n) for n=0..2401
Index entries for the Kaprekar map

Cf. A165071, A165073.

In other bases: A164886 (base 2), A164996 (base 3), A165015 (base 4), A165035 (base 5), A165054 (base 6), A165093 (base 8), A165113 (base 9), A151963 (base 10).

cp's OEIS Frontend

A163205 The non-repetitive Kaprekar binary numbers in decimal.

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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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A164885 Length of preperiodic part of trajectory of n under iteration of the base-2 Kaprekar map in A164884.

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A164887 a(n) = smallest number that leads to a new fixed point under the base-2 Kaprekar map of A164884.

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A151963 (Length of preperiodic part) + (length of cycle) of trajectory of n under iteration of the Kaprekar map in A151949.

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Maple

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A164996 (Length of preperiodic part) + (length of cycle) of trajectory of n under iteration of the base-3 Kaprekar map in A164993.

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A165015 (Length of preperiodic part) + (length of cycle) of trajectory of n under iteration of the base-4 Kaprekar map in A165012.

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A165035 (Length of preperiodic part) + (length of cycle) of trajectory of n under iteration of the base-5 Kaprekar map in A165032.

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A165054 (Length of preperiodic part) + (length of cycle) of trajectory of n under iteration of the base-6 Kaprekar map in A165051.

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A165074 (Length of preperiodic part) + (length of cycle) of trajectory of n under iteration of the base-7 Kaprekar map in A165071.

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