1,2

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 (vincenzo.librandi(AT)tin.it), Jan 05 2009]

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

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

# 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

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 this_sequence A055160 A055157 A027808

Adjacent sequences: A099006 A099007 A099008 this_sequence A099010 A099011 A099012

nonn,base

Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Sep 22 2004

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