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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. 18
0, 53955, 64308654, 61974, 86420987532 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

No cycle of length 6 is presently known!

It is also known that a(7) = 420876, a(8) = 7509843, a(14) = 753098643.

Comments from Joseph Myers, Aug 19 2009 (Start)

One does not to need to consider every integer of n digits, only the sorted

sequences of n digits (of which there are n+9 choose 9, so 28048800 for 23

digits). Then you only need to consider those sorted sequences of digits

whose total is a multiple of 9, as the number and so the sum of its digits

is always a multiple of 9 after the first iteration, which reduces the

work by a further factor of about 9.

As a further refinement, the result of a single subtraction, if not zero, will have digit sequence of the form

d_1 d_2 ... d_k-1 9...9 9-d_k ... 9-d_2 9-d_1+1

where the values d_i are in the range 1 to 9 and the sequence of 9s in the middle may be empty.

From this form it follows that for any member of a cycle,

abs(number of 8s - number of 1s) + abs(number of 7s - number of 2s) +

abs(number of 6s - number of 3s) + abs(number of 5s - number of 4s) +

max(0, number of 0s - number of 9s) <= 4,

so given the numbers of 0s, 1s, 2s, 3s and 4s there is little freedom left

in choosing the number of each remaining digit.

No further cycle lengths exist up to at least 140 digits. The only 4-cycles up

to there are the ones containing 61974 and 62964, the only 8-cycles up to there are

the ones containing 7509843 and 76320987633, the only 14-cycle up to there

is the one containing 753098643. All the 7-cycles so far follow the pattern

7-cycle: 420876

7-cycle: 43208766

7-cycle: 4332087666

7-cycle: 433320876666

7-cycle: 43333208766666

7-cycle: 4333332087666666 ... (End)

LINKS

Table of n, a(n) for n=1..5.

R. J. Mathar, Maple code for A151949 and A151959

Joseph Myers, List of cycles under Kaprekar map (all numbers with <= 60 digits; cycles are represented by their smallest value)

Index entries for the Kaprekar map

EXAMPLE

a(1) = 0: 0 -> 0.

a(2) = 53955: 53955 -> 59994 -> 53955 -> ...

a(3) = 64308654: 64308654 -> 83208762 -> 86526432 -> 64308654 -> ...

a(4) = 61974: 61974 -> 82962 -> 75933 -> 63954 -> 61974 -> ...

CROSSREFS

A099009 gives the fixed points and A099010 gives numbers in cycles of length > 1.

Cf. A151949.

In other bases: A153881 (base 2), A165008 (base 3), A165028 (base 4), A165047 (base 5), A165067 (base 6), A165086 (base 7), A165106 (base 8), A165126 (base 9). [From Joseph Myers, Sep 05 2009]

Sequence in context: A099010 A164723 A164720 * A164724 A190472 A038564

Adjacent sequences:  A151956 A151957 A151958 * A151960 A151961 A151962

KEYWORD

nonn,more,base

AUTHOR

Klaus Brockhaus and N. J. A. Sloane, Aug 19 2009

EXTENSIONS

The term a(3) = 64308654 was initially only a conjecture, but was confirmed by Zak Seidov, Aug 19 2009

a(4) = 61974 corrected by R. J. Mathar, Aug 19 2009 (we had not given the smallest member of the 4-cycle).

a(4) = 61974 confirmed by Zak Seidov, Aug 19 2009

a(7) = 420876 confirmed by Zak Seidov, Aug 19 2009

a(8) = 7509843 confirmed by Zak Seidov, Aug 19 2009 (formerly this was just an upper bound)

a(5) = 86420987532 and a(14) = 753098643 from Joseph Myers, Aug 19 2009. He also confirms the other values, and remarks that there are no other cycle lengths up to at least 140 digits.

STATUS

approved

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Last modified August 20 06:32 EDT 2017. Contains 290824 sequences.