|
| |
|
|
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.
|
|
31
| |
|
|
0, 495, 6174, 549945, 631764, 63317664, 97508421, 554999445, 864197532, 6333176664, 9753086421, 9975084201, 86431976532, 555499994445, 633331766664, 975330866421, 997530864201, 999750842001, 8643319766532, 63333317666664
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 1,2
|
|
|
COMMENTS
| There are no seven-digit 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 (jens.k.a(AT)get2net.dk), 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. [From Vincenzo Librandi, Nov 17 2010]
|
|
|
LINKS
| Joseph Myers, Table of n, a(n) for n=1..5344
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, KaprekarRoutine
Index entries for the Kaprekar map
|
|
|
EXAMPLE
| 6174 is a fixed point of the mapping and hence a term: 6174 -> 7641 - 1467 = 6174.
|
|
|
PROG
| # Python (2.4) program 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, n-1):
........... 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
|
|
|
CROSSREFS
| Cf. A090429, A069746, A099010, A151959.
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]
Sequence in context: A164718 A151965 A151957 * A055160 A055157 A027808
Adjacent sequences: A099006 A099007 A099008 * A099010 A099011 A099012
|
|
|
KEYWORD
| nonn,base
|
|
|
AUTHOR
| Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Sep 22 2004
|
|
|
EXTENSIONS
| More terms from Jens Kruse Andersen (jens.k.a(AT)get2net.dk) and Tim Peters (tim(AT)python.org), Oct 04 2004
Corrected by Jens Kruse Andersen (jens.k.a(AT)get2net.dk), Oct 25 2004
|
| |
|
|
