

A099009


Fixed points of the Kaprekar mapping f(n) = n'  n'', where in n' the digits of n are arranged in descending, in n'' in ascending order.


42



0, 495, 6174, 549945, 631764, 63317664, 97508421, 554999445, 864197532, 6333176664, 9753086421, 9975084201, 86431976532, 555499994445, 633331766664, 975330866421, 997530864201, 999750842001, 8643319766532, 63333317666664
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OFFSET

1,2


COMMENTS

There are no sevendigit fixed points.
Let d(n) denote n repetitions of the digit d. The sequence includes the following for all n>=0: 5(n)499(n)4(n)5, 63(n)176(n)4, 8643(n)1976(n)532.  Jens Kruse Andersen, Oct 04 2004
0's in n giving leading 0's in n'' is allowed.
For every natural number n let n' and n" be the numbers obtained by arranging the digits of n into decreasing and increasing order, and let f(n)=n'n". It is known that the number 6174 is invariant under this transformation and that applying f a certain number of times to a number n with four digits the numbers 0, 495 or 6174 are always reached.  Vincenzo Librandi, Nov 17 2010
Each term of A055162(n) corresponds to A099009(n+1), with its digits being reordered in the ascending manner.  Alexander R. Povolotsky, Apr 27 2012
All terms of this sequence are divisible by nine, a(n)/9 = A132155(n).  Alexander R. Povolotsky, Apr 29 2012
A055160 differs from this sequence only at the positions of two terms in it: 554999445 and 555499994445.  Alexander R. Povolotsky, May 01 2012
The union of the sequences A214555, A214556, A214557, A214558, A214559 and the element 0 gives the sequence A099009.  Syed Iddi Hasan, Jul 24 2012
The comment made by Jens Kruse Andersen is missing one more family of terms (which starts with one or more digits "9" and ends with the digit "1"): 97508421, 9753086421, 9975084201, 975330866421, 997530864201, 999750842001, ... This family could be generalized (using the same method as in Andersen's comment) and it is actually covered by Syed Iddi Hasan in A214559. Also A214557 and A214558 (both  by Syed Iddi Hasan) are variants of Andersen's 8643(n)1976(n)532.  Alexander R. Povolotsky, Mar 14 2015
Fixed points of A151949.  Reinhard Zumkeller, Mar 23 2015


LINKS

Syed Iddi Hasan, Table of n, a(n) for n = 1..8924
Mauro Fiorentini, Kaprekar (costante di) (in Italian)
Manuj Mishra, Illustration of first 8923 terms, with each digit in a different color
Manuj Mishra, Illustration as above but only including terms of even length
Manuj Mishra, Illustration as above but only including terms of odd length
Joseph Myers, List of cycles under Kaprekar map (all numbers with <= 60 digits; cycles are represented by their smallest value)
Conrad Roche, Kaprekar Series Generator.
Eric Weisstein's World of Mathematics, Kaprekar Routine
Index entries for the Kaprekar map


EXAMPLE

6174 is a fixed point of the mapping and hence a term: 6174 > 7641  1467 = 6174.


MATHEMATICA

f[n_] := Block[{d = IntegerDigits@ n, a, b}, a = FromDigits@ Sort@ d; b = FromDigits@ Reverse@ Sort@ d; n == b  a]; Select[Range@ 1000000, f] (* Michael De Vlieger, Mar 20 2015 *)


PROG

(Python)
# (version 2.4) from Tim Peters (Replace leading dots by blanks before running)
.def extend(base, start, n):
... if n == 0:
....... yield base
....... return
... for i in range(start, 10):
....... for x in extend(base + str(i), i, n1):
........... yield x
.def drive(n):
... result = []
... for lo in extend("", 0, n):
....... ilo = int(lo)
....... if ilo == 0 and n > 1:
........... continue
....... hi = lo[::1]
....... diff = str(int(hi)  ilo)
....... diff = "0" * (n  len(diff)) + diff
....... if sorted(diff) == list(lo):
........... result.append(diff)
... return sorted(result)
.for n in range(1, 17):
... print "Length", n
... print '' * 40
... for r in drive(n):
....... print r
(Haskell)
a099009 n = a099009_list !! (n1)
a099009_list = [x  x < [0..], a151949 x == x]
 Reinhard Zumkeller, Mar 23 2015
(MAGMA) a:=func<nSeqint(Sort(Intseq(n)))Seqint(Reverse(Sort(Intseq(n)))) eq n>; [k:k in [0..10^7]a(k)]; // Marius A. Burtea, Sep 12 2019


CROSSREFS

Cf. A090429, A069746, A099010, A151959, A055162, A132155, A055160, A214557, A214558, A214559
In other bases: A163205 (base 2), A164997 (base 3), A165016 (base 4), A165036 (base 5), A165055 (base 6), A165075 (base 7), A165094 (base 8), A165114 (base 9).
Cf. A151949, A004185, A004186.
Sequence in context: A164718 A151965 A151957 * A055160 A055157 A027808
Adjacent sequences: A099006 A099007 A099008 * A099010 A099011 A099012


KEYWORD

nonn,base


AUTHOR

Klaus Brockhaus, Sep 22 2004


EXTENSIONS

More terms from Jens Kruse Andersen and Tim Peters (tim(AT)python.org), Oct 04 2004
Corrected by Jens Kruse Andersen, Oct 25 2004


STATUS

approved



