# [seqfan] help needed with sequences related to Kaprekar map

N. J. A. Sloane njas at research.att.com
Wed Aug 19 19:19:15 CEST 2009

```Dear Sequence Fans,  Now that the OEIS is "on vacation" I have
time to read the newspaper.  Yesterday's New York Times (Science Section,
Aug 18 2009, last page) has three sequences as puzzles. Two of
them were in the OEIS, the third was not (it is now A151946).

The rule for the third sequence is the Kaprekar map, see A151949, given by

n -> K(n) := (n with digits sorted into descending order) - (n with digits sorted into ascending order)

E.g. K(102) = 210 - (0)12 = 210 - 12 = 198.

With help from Klaus Brockhaus and Harvey Dale, I have added many new sequences
related to this map, and there is also an Index entry.  The sequences related to
this map are presently:

A151949*, A099009*, A099010, A069746, A090429, A132155, A160761, A151946, A151947, A151950, A056965, A151951, A151955, A151956, A151957, A151958, A151959, A151962, A151963, A151964, A151965, A151966

(Klaus's A099009 gives the fixed points)

I am writing to ask the sequence fans for help in extending these sequences - many
of them need more terms.

The most important outstanding question concerns the smallest cycle of length 3
- is it   64308654 -> 83208762 -> 86526432 -> 64308654 ...  or is there a smaller example?

In other words, the third term of the following entry needs to be confirmed or corrected!

%I A151959
%S A151959 0,53955,64308654,62964
%N A151959 Consider the Kaprekar map x->K(x) described in A151949. Sequence gives the smallest number that belongs to a cycle of length n under repeated iteration of this map, or -1 if there is no cycle of length n.
%C A151959 The term a(3) = 64308654 is only a conjecture, and needs to be confirmed.
%C A151959 No cycles of lengths 5 0r 6 are presently known.
%C A151959 It is also known that a(7) = 420876 and a(8) <= 7509843.
%C A151959 A099009 gives the fixed points and A099010 gives numbers in cycles of length > 1.
%H A151959 <a href="Sindx_K.html#Kaprekar_map">Index entries for the Kaprekar map</a>
%e A151959 a(1) = 0: 0 -> 0.
%e A151959 a(2) = 53955: 53955 -> 59994 -> 53955 -> ...
%e A151959 a(3) = 64308654?: 64308654 -> 83208762 -> 86526432 -> 64308654 -> 83208762 -> ..., but there is a possibilty that a smaller example exists.
%O A151959 0,2
%K A151959 nonn,more
%A A151959 K. Brockhaus (klaus-brockhaus(AT)t-online.de) and N. J. A. Sloane (njas(AT)research.att.com), Aug 19 2009

But many others need extending too.

Neil

```