[seqfan] Re: help needed with sequences related to Kaprekar map

[zak seidov]

Neil,
I confirm that, in A151959, a(3) = 64308654 
(no smaller number with cycle of length 3).
I used brute force with Mathematica.
Zak



--- On Wed, 8/19/09, N. J. A. Sloane <njas at research.att.com> wrote:

> From: N. J. A. Sloane <njas at research.att.com>
> Subject: [seqfan]  help needed with sequences related to Kaprekar map
> To: seqfan at seqfan.eu
> Cc: njas at research.att.com
> Date: Wednesday, August 19, 2009, 1:19 PM
> 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
> 
> 
> 
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