A099009 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.

0, 495, 6174, 549945, 631764, 63317664, 97508421, 554999445, 864197532, 6333176664, 9753086421, 9975084201, 86431976532, 555499994445, 633331766664, 975330866421, 997530864201, 999750842001, 8643319766532, 63333317666664

Offset: 1

Author

Klaus Brockhaus, Sep 22 2004

Keywords

• nonn
• base

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, 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. - Vincenzo Librandi, Nov 17 2010

Each term of A055162(n) corresponds to A099009(n+1), with its digits being reordered in the ascending manner. - Alexander R. Povolotsky, Apr 27 2012

All terms of this sequence are divisible by nine, a(n)/9 = A132155(n). - Alexander R. Povolotsky, Apr 29 2012

A055160 differs from this sequence only at the positions of two terms in it: 554999445 and 555499994445. - Alexander R. Povolotsky, May 01 2012

The union of the sequences A214555, A214556, A214557, A214558, A214559 and the element 0 gives the sequence A099009. - Syed Iddi Hasan, Jul 24 2012

The comment made by Jens Kruse Andersen is missing one more family of terms (which starts with one or more digits "9" and ends with the digit "1"): 97508421, 9753086421, 9975084201, 975330866421, 997530864201, 999750842001, ... This family could be generalized (using the same method as in Andersen's comment) and it is actually covered by Syed Iddi Hasan in A214559. Also A214557 and A214558 (both - by Syed Iddi Hasan) are variants of Andersen's 8643(n)1976(n)532. - Alexander R. Povolotsky, Mar 14 2015

Fixed points of A151949. - Reinhard Zumkeller, Mar 23 2015

Examples

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

Links

• Syed Iddi Hasan, Table of n, a(n) for n = 1..8924
• Mauro Fiorentini, Kaprekar (costante di) (in Italian)
• Manuj Mishra, Illustration of first 8923 terms, with each digit in a different color
• Manuj Mishra, Illustration as above but only including terms of even length
• Manuj Mishra, Illustration as above but only including terms of odd length
• 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, Kaprekar Routine
• Index entries for the Kaprekar map

Crossrefs

Cf. A090429, A069746, A099010, A151959, A055162, A132155, A055160, A214557, A214558, A214559

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).

Cf. A151949, A004185, A004186.

Programs

Haskell

a099009 n = a099009_list !! (n-1)
a099009_list = [x | x <- [0..], a151949 x == x]
-- Reinhard Zumkeller, Mar 23 2015

Magma

a:=func; [k:k in [0..10^7]|a(k)]; // Marius A. Burtea, Sep 12 2019

Mathematica

f[n_] := Block[{d = IntegerDigits@ n, a, b}, a = FromDigits@ Sort@ d; b = FromDigits@ Reverse@ Sort@ d; n == b - a]; Select[Range@ 1000000, f] (* Michael De Vlieger, Mar 20 2015 *)

Python

# (version 2.4) from Tim Peters
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, end=', ')

Extensions

More terms from Jens Kruse Andersen and Tim Peters (tim(AT)python.org), Oct 04 2004

Corrected by Jens Kruse Andersen, Oct 25 2004