%I A099009
%S A099009 0,495,6174,549945,631764,63317664,97508421,554999445,864197532,
%T A099009 6333176664,9753086421,9975084201,86431976532,555499994445,633331766664,
%U A099009 975330866421,997530864201,999750842001,8643319766532,63333317666664
%N A099009 List of 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.
%C A099009 There are no seven-digit fixed points.
%C A099009 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 (jens.k.a(AT)get2net.dk), Oct 04 2004
%C A099009 0's in n giving leading 0's in n'' is allowed.
%C A099009 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. [From 
               Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Jan 05 2009]
%H A099009 Joseph Myers, <a href="b099009.txt">Table of n, a(n) for n=1..5344</a>
%H A099009 Joseph Myers, <a href="a099009.txt">List of cycles under Kaprekar map</
               a> (all numbers with <= 60 digits; cycles are represented by their 
               smallest value)
%H A099009 Conrad Roche, <a href="http://kaprekar.sourceforge.net">Kaprekar Series 
               Generator</a>.
%H A099009 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               KaprekarRoutine.html">KaprekarRoutine</a>
%H A099009 <a href="Sindx_K.html#Kaprekar_map">Index entries for the Kaprekar map</
               a>
%e A099009 6174 is a fixed point of the mapping and hence a term: 6174 -> 7641 - 
               1467 = 6174.
%o A099009 # Python (2.4) program from Tim Peters (Replace leading dots by blanks 
               before running)
%o A099009 .def extend(base, start, n):
%o A099009 ... if n == 0:
%o A099009 ....... yield base
%o A099009 ....... return
%o A099009 ... for i in range(start, 10):
%o A099009 ....... for x in extend(base + str(i), i, n-1):
%o A099009 ........... yield x
%o A099009 .def drive(n):
%o A099009 ... result = []
%o A099009 ... for lo in extend("", 0, n):
%o A099009 ....... ilo = int(lo)
%o A099009 ....... if ilo == 0 and n > 1:
%o A099009 ........... continue
%o A099009 ....... hi = lo[::-1]
%o A099009 ....... diff = str(int(hi) - ilo)
%o A099009 ....... diff = "0" * (n - len(diff)) + diff
%o A099009 ....... if sorted(diff) == list(lo):
%o A099009 ........... result.append(diff)
%o A099009 ... return sorted(result)
%o A099009 .for n in range(1, 17):
%o A099009 ... print "Length", n
%o A099009 ... print '-' * 40
%o A099009 ... for r in drive(n):
%o A099009 ....... print r
%Y A099009 Cf. A090429, A069746, A099010, A151959.
%Y A099009 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). [From Joseph Myers (jsm(AT)polyomino.org.uk), 
               Sep 05 2009]
%Y A099009 Sequence in context: A164718 A151965 A151957 this_sequence A055160 A055157 
               A027808
%Y A099009 Adjacent sequences: A099006 A099007 A099008 this_sequence A099010 A099011 
               A099012
%K A099009 nonn,base
%O A099009 1,2
%A A099009 Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Sep 22 2004
%E A099009 More terms from Jens Kruse Andersen (jens.k.a(AT)get2net.dk) and Tim 
               Peters (tim(AT)python.org), Oct 04 2004
%E A099009 Corrected by Jens Kruse Andersen (jens.k.a(AT)get2net.dk), Oct 25 2004