A004186 -id:A004186 - cp's OEIS frontend

A352544 a(n) = n/2 if n is even, otherwise n + A004186(n), where A004186 = arrange digits in decreasing order.

0, 2, 1, 6, 2, 10, 3, 14, 4, 18, 5, 22, 6, 44, 7, 66, 8, 88, 9, 110, 10, 42, 11, 55, 12, 77, 13, 99, 14, 121, 15, 62, 16, 66, 17, 88, 18, 110, 19, 132, 20, 82, 21, 86, 22, 99, 23, 121, 24, 143, 25, 102, 26, 106, 27, 110, 28, 132, 29, 154, 30, 122, 31, 126, 32, 130, 33, 143, 34, 165, 35, 142, 36, 146, 37, 150, 38, 154, 39, 176, 40, 162, 41, 166, 42, 170, 43, 174, 44, 187, 45

Offset: 0

Eric Angelini and M. F. Hasler, Mar 20 2022

A variant of the Collatz (3x+1) map A006370. See A352540 - A352543 for more about iterations of this map.

Eric Angelini, Divide by 2 or add the biggest anagram, math-fun discussion list, Mar 20 2022

Cf. A004186.
Coincides with A064680 (half if even, double if odd) for n < 13.
Cf. A352540 (initial values with infinite orbit under A352544), A352541 (number of iterations of A352544 to reach a value for the second time), A352542 (orbit of 89 under A352544), A352543 (numbers that end in a loop of length > 3 under A352544), A352545 (representatives of loops of length > 3).

apply( {A352544(n)=if(n%2,n+A004186(n),n\2)}, [0..90]) \\ with A004186(n)=fromdigits(vecsort(digits(n),,4))

A096091 Numbers n with property that largest number formed from digits of n (A004186(n)) is divisible by smallest number formed from digits of n (A004185(n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 20, 22, 30, 33, 40, 44, 50, 55, 60, 66, 70, 77, 80, 88, 90, 99, 100, 101, 105, 108, 110, 111, 150, 180, 200, 202, 220, 222, 300, 303, 330, 333, 400, 404, 405, 440, 444, 450, 500, 501, 504, 505, 510, 540, 550, 555, 600, 606

Offset: 1

Amarnath Murthy, Jun 22 2004

The number N = d*10^m*(10^k-1)/9 is a member for all m, k where 1 < d < 10, since the quotient is 10^m. E.g., for d = 7, m = 4, k = 8 we get N = 777777770000.
Conjecture: There are infinitely many terms besides these.
From Jon E. Schoenfield, Jul 26 2015: (Start)
Every number whose nonzero digits are all identical (e.g., 70770070777) is a term in the sequence (so the sequence is infinite). Also, if k is a term, then so is k*10 (hence, so is k*10^m for m >= 1). Removal of all terms that satisfy either of the above criteria still leaves an infinite number of terms, beginning with 105, 108, 150, 180, 405, 450, 501, 504, 510, 540, 801, 810, ... (see A260461).
If any integer k is a term, then so is every integer obtained by permuting the digits of k, except for some (not necessarily all) permutations beginning with 0. E.g., since 12000 is a member, so are all the other permutations of its digits that begin with 1 (i.e., 10002, 10020, and 10200), and all those that begin with 2 (i.e., 20001, 20010, 20100, and 21000), as well as the permutations that begin with a single 0 (which, after leading zeros are removed, reduce to 1002, 1020, 1200, 2001, 2010, and 2100), but not those that begin with more than one 0 (i.e., the sequence does not include 12, 21, 102, 120, 201, or 210). Aside from those terms whose nonzero digits are all identical, it appears that only a small number of patterns result from sorting the digits in increasing order (and discarding the zeros, which, of course, are all leading zeros): these "primitives" begin with 12, 15, 16, 18, 24, 25, 36, 45, 48, 125, ... (see A260462). (End)

			110 is a member as 110/011=10.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A004185, A004186, A096089, A096090, A260461, A260462.

isA096091 := proc(n)
    if modp( A004186(n),A004185(n))= 0 then
        true;
    else
        false;
    end if;
end proc:
for n from 1 to 1000 do
    if isA096091(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Jul 26 2015

Select[Range[999], (d = Sort@ IntegerDigits@ #; Divisible@@ FromDigits/@ {Reverse@ d, d})&] (* Giovanni Resta, Jul 26 2015 *)

Edited, corrected and extended by Robert G. Wilson v, Jun 23 2004

Entry revised by Editors of the OEIS, Jul 26 2015

A283001 a(n) = (A004186(n) - n)/9.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 0, 0, 0, 1, 2, 3, 4, 5, 6, 7, 0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 0

David A. Corneth, Feb 26 2017

A004186(n) is the number formed when the digits of n are sorted nonincreasingly.

Is this the same as A037887 or A037851? - R. J. Mathar, Mar 06 2017

			a(141) = (411 - 141)/9 = 30.

David A. Corneth, Table of n, a(n) for n = 0..9999

Cf. A004186, A037887, A037851.

s[n_]:= FromDigits@ Reverse@ Sort@ IntegerDigits@ n; a[n_]:=(s[n]-n)/9; Table[a[n],{n,0,89}] (* Indranil Ghosh, Feb 26 2017 *)

a(n) = my(d = digits(n), v=vecsort(d,,4)-d); sum(i=1,#v,v[i]*10^(#v-i)) / 9

def A283001(n): return (int("".join(sorted(str(n),reverse=True)))-n)/9 # Indranil Ghosh, Feb 26 2017

a(10*n) = 10*a(n).

A337171 a(n) = A004186(n) mod n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 9, 5, 13, 6, 13, 3, 9, 15, 0, 0, 0, 9, 18, 2, 10, 18, 26, 5, 0, 0, 0, 0, 9, 18, 27, 36, 7, 15, 0, 0, 0, 0, 0, 9, 18, 27, 36, 45, 0, 0, 0, 0, 0, 0, 9, 18, 27, 36, 0, 0, 0, 0, 0, 0, 0, 9, 18, 27, 0, 0, 0, 0, 0, 0, 0, 0, 9, 18

Offset: 1

Simon Strandgaard, Jan 28 2021

			a(12) = A004186(12) mod 12 = 21 mod 12 = 9,
a(13) = A004186(13) mod 13 = 31 mod 13 = 5,
a(14) = A004186(14) mod 14 = 41 mod 14 = 13,
a(15) = A004186(15) mod 15 = 51 mod 15 = 6,
a(16) = A004186(16) mod 16 = 61 mod 16 = 13.

Simon Strandgaard, Plot of 1000 terms

Cf. A004186.

Cf. A319650 (similar in binary).

a[n_] := Mod[FromDigits @ Sort[IntegerDigits[n], Greater], n]; Array[a, 100] (* Amiram Eldar, Jan 30 2021 *)

a(n) = fromdigits(vecsort(digits(n), , 4)) % n; \\ Michel Marcus, Mar 04 2021

values = (1..20).map do |n|
    n.to_s.split('').sort.join.reverse.to_i % n
end
p values

A004086 Read n backwards (referred to as R(n) in many sequences).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 11, 21, 31, 41, 51, 61, 71, 81, 91, 2, 12, 22, 32, 42, 52, 62, 72, 82, 92, 3, 13, 23, 33, 43, 53, 63, 73, 83, 93, 4, 14, 24, 34, 44, 54, 64, 74, 84, 94, 5, 15, 25, 35, 45, 55, 65, 75, 85, 95, 6, 16, 26, 36, 46, 56, 66, 76, 86, 96, 7, 17, 27, 37, 47

Offset: 0

N. J. A. Sloane

Also called digit reversal of n.

Leading zeros (after the reversal has taken place) are omitted. - N. J. A. Sloane, Jan 23 2017

Indranil Ghosh, Table of n, a(n) for n = 0..50000 (first 1001 terms from Franklin T. Adams-Watters)
Dana G. Korssjoen, Biyao Li, Stefan Steinerberger, Raghavendra Tripathi, and Ruimin Zhang, Finding structure in sequences of real numbers via graph theory: a problem list, arXiv:2012.04625 [math.CO], Dec 08 2020.
Michael Penn, A digit moving number puzzle., YouTube video, 2022.

Cf. A004185, A004186, A009996, A073138, A188649, A221714.

Cf. A030101, A030102, A030103, A030104, A030105, A030107, A030108.

a004086 = read . reverse . show  -- Reinhard Zumkeller, Apr 11 2011

|.&.": i.@- 1e5 NB. Stephen Makdisi, May 14 2018

read transforms; A004086 := digrev; #cf "Transforms" link at bottom of page
A004086:=proc(n) local s,t; if n<10 then n else s:=irem(n,10,'t'); while t>9 do s:=s*10+irem(t,10,'t') od: s*10+t fi end; # M. F. Hasler, Jan 29 2012

Table[FromDigits[Reverse[IntegerDigits[n]]], {n, 0, 75}]
IntegerReverse[Range[0,80]](* Requires Mathematica version 10 or later *) (* Harvey P. Dale, May 13 2018 *)

dig(n) = {local(m=n,r=[]); while(m>0,r=concat(m%10,r);m=floor(m/10));r}
A004086(n) = {local(b,m,r);r=0;b=1;m=dig(n);for(i=1,matsize(m)[2],r=r+b*m[i];b=b*10);r} \\ Michael B. Porter, Oct 16 2009

A004086(n)=fromdigits(Vecrev(digits(n))) \\ M. F. Hasler, Nov 11 2010, updated May 11 2015, Sep 13 2019

def A004086(n):
    return int(str(n)[::-1]) # Chai Wah Wu, Aug 30 2014

For n > 0, a(a(n)) = n iff n mod 10 != 0. - Reinhard Zumkeller, Mar 10 2002
a(n) = d(n,0) with d(n,r) = r if n=0, otherwise d(floor(n/10), r*10+(n mod 10)). - Reinhard Zumkeller, Mar 04 2010
a(10*n+x) = x*10^m + a(n) if 10^(m-1) <= n < 10^m and 0 <= x <= 9. - Robert Israel, Jun 11 2015

Extended by Ray Chandler, Dec 30 2004

A151949 a(n) = image of n under the Kaprekar map n -> (n with digits sorted into descending order) - (n with digits sorted into ascending order).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 9, 0, 9, 18, 27, 36, 45, 54, 63, 72, 18, 9, 0, 9, 18, 27, 36, 45, 54, 63, 27, 18, 9, 0, 9, 18, 27, 36, 45, 54, 36, 27, 18, 9, 0, 9, 18, 27, 36, 45, 45, 36, 27, 18, 9, 0, 9, 18, 27, 36, 54, 45, 36, 27, 18, 9, 0, 9, 18, 27, 63, 54, 45, 36, 27, 18, 9, 0, 9, 18, 72, 63, 54, 45, 36, 27, 18, 9, 0, 9, 81, 72, 63, 54, 45, 36, 27, 18, 9, 0, 99, 99, 198, 297, 396, 495, 594, 693, 792, 891, 99, 0, 99, 198, 297, 396, 495, 594, 693, 792

Offset: 0

N. J. A. Sloane, Aug 18 2009

Entries are multiples of 9 - see A151950.

a(n) = A004186(n) - A004185(n); a(A010785(n)) = 0. - Reinhard Zumkeller, corrected: Mar 23 2015, Jul 09 2013

			For n = 15, a(15) = 51 - 15 = 36. - _Indranil Ghosh_, Feb 01 2017

Indranil Ghosh, Table of n, a(n) for n = 0..20000 (terms 0..1000 from Joseph Myers and Robert G. Wilson v)
H. Hanslik, E. Hetmaniok, I. Sobstyl, et al., Orbits of the Kaprekar's transformations-some introductory facts, Zeszyty Naukowe Politechniki Śląskiej, Seria: Matematyka Stosowana z. 5, Nr kol. 1945; 2015.
M. Kauers and C. Koutschan, Some D-finite and some possibly D-finite sequences in the OEIS, arXiv:2303.02793 [cs.SC], 2023.
R. J. Mathar, Maple code for A151949 and A151959
Joseph Myers, C program for computing sequences related to the Kaprekar map
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

In other bases: A164884 (base 2), A164993 (base 3), A165012 (base 4), A165032 (base 5), A165051 (base 6), A165071 (base 7), A165090 (base 8), A165110 (base 9). - Joseph Myers, Sep 05 2009
Cf. A151959, A069746, A347688.
Cf. also A004185, A004186, A099009 (fixed points).

a151949 n = a004186 n - a004185 n
-- Reinhard Zumkeller, corrected: Mar 23 2015, Jul 09 2013

f[n_] := Module[{idn = IntegerDigits@n, idns}, idns = Sort@ idn; FromDigits@ Reverse@ idns - FromDigits@ idns]; Table[ f@n, {n, 0, 200}] (* Harvey P. Dale, Aug 18 2009 *)
Flatten[Table[Differences[FromDigits /@ {y = Sort[x = IntegerDigits[n]], Reverse[y]}], {n, 0, 74}]] (* Jayanta Basu, Jul 11 2013 *)

a(n) = {my(d=digits(n)); fromdigits(vecsort(d,,4)) - fromdigits(vecsort(d));} \\ Michel Marcus, Dec 08 2019

def A151949(n):
    return int("".join(sorted(str(n),reverse=True)))-int("".join(sorted(str(n)))) # Indranil Ghosh, Feb 01 2017

More terms from Robert G. Wilson v, Aug 19 2009

More than the usual number of terms are shown in order to distinguish this from similar sequences. - N. J. A. Sloane, Sep 22 2021

A004185 Arrange digits of n in increasing order, then (for n > 0) omit the zeros.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 11, 12, 13, 14, 15, 16, 17, 18, 19, 2, 12, 22, 23, 24, 25, 26, 27, 28, 29, 3, 13, 23, 33, 34, 35, 36, 37, 38, 39, 4, 14, 24, 34, 44, 45, 46, 47, 48, 49, 5, 15, 25, 35, 45, 55, 56, 57, 58, 59, 6, 16, 26, 36, 46, 56, 66, 67, 68, 69, 7, 17, 27, 37, 47

Offset: 0

N. J. A. Sloane

Record values: A009994. - Reinhard Zumkeller, Dec 05 2009
If we define "sortable primes" as prime numbers that remain prime when their digits are sorted in increasing order, then all absolute primes (A003459) are sortable primes but not all sortable primes are absolute primes. For example, 311 is both sortable and absolute, and 271 is sortable but not absolute, since its digits can be permuted to 217 = 7 * 31 or 712 = 2^3 * 89, etc. - Alonso del Arte, Oct 05 2013
The above mentioned "sortable primes" are listed in A211654, the nontrivial ones (with digits not in nondecreasing order) in A086042. - M. F. Hasler, Jul 30 2019

			a(19) = 19 because the digits are already in increasing order.
a(20) = 2 because the digits of 20 are 2 and 0, which in increasing order are 0 and 2, but since zero-padding is not allowed on the left, the zero digit is dropped and we are left with 2.
a(21) = 12 because the digits of 21 are 2 and 1, which in increasing order are 1 and 2.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A004719, A004151, A004086, A004186, A009996, A221714, A073138, A193581, A193582, A065641.

Cf. A211654 (sortable primes) and subsequence A086042 (nontrivial solutions).

import Data.List (sort)
a004185 n = read $ sort $ show n :: Integer
-- Reinhard Zumkeller, Aug 10 2011

A004185:=func; [n eq 0 select 0 else A004185(n): n in [0..57]]; // Bruno Berselli, Apr 03 2012

A004185 := proc(n)
    local dgs;
    convert(n,base,10) ;
    dgs := sort(%,`>`) ;
    add( op(i,dgs)*10^(i-1),i=1..nops(dgs)) ;
end proc:
seq(A004185(n),n=0..20) ; # R. J. Mathar, Jul 26 2015

FromDigits[Sort[DeleteCases[IntegerDigits[#], 0]]]&/@Range[0, 60] (* Harvey P. Dale, Nov 29 2011 *)

a(n)=fromdigits(vecsort(digits(n))) \\ Charles R Greathouse IV, Feb 06 2017

def A004185(n):
    return int(''.join(sorted(str(n))).replace('0','')) if n > 0 else 0 # Chai Wah Wu, Nov 10 2015

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.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Sep 22 2004

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

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

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

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.

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

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

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 *)

# (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=', ')

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

Corrected by Jens Kruse Andersen, Oct 25 2004

A179239 Permutation classes of integers, each identified by its smallest member.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27, 28, 29, 30, 33, 34, 35, 36, 37, 38, 39, 40, 44, 45, 46, 47, 48, 49, 50, 55, 56, 57, 58, 59, 60, 66, 67, 68, 69, 70, 77, 78, 79, 80, 88, 89, 90, 99, 100, 101, 102, 103

Offset: 0

Aaron Dunigan AtLee, Jul 04 2010

Let the "permutation set" of a positive integer n be the set of all integers formed by permuting the digits of n. Two integers are "permutationally congruent" if they generate the same permutation set. A "permutation class" is a set of all permutationally congruent integers. This sequence lists each permutation class, identified by its smallest member.
These are also the positive integers in order, omitting any d-digit number n if a previously listed d-digit number is a permutation of the digits of n.
Range of A328447: smallest representative of the equivalence class of all numbers having the same digits up to permutation. Equivalently: Numbers with digits in nondecreasing order, except that the smallest nonzero digit must precede the zero digits. This sequence is useful when considering functions which depend only on the digits of n, e.g., the number of primes contained in n, cf. A039993, A039999, A075053 and the records therein, A072857 (primeval numbers) and A076497, resp. A239196 and A239197, etc. - M. F. Hasler, Oct 18 2019

			The permutation set of 24 is {24, 42}, and this is the equivalence class modulo permutations of both of them, so 24 is listed, but 42 is not.
The permutation set of 30 is {3, 30}, but 3 is not in the same permutation class as 30 since 30 cannot be obtained by permuting digits of 3. Therefore 30 is listed separately from 3.
The numbers 89 and 98 are also permutationally congruent and form a permutation class, so only the smaller one is listed.

Aaron Dunigan AtLee and Michael De Vlieger, Table of n, a(n) for n = 0..10000 (first 2997 terms from Aaron Dunigan AtLee; prefix 0 by _Georg Fischer_, Oct 24 2019)

A variant of A009994.
Cf. A047726, A035927 (Number of distinct n-digit numbers up to permutations of digits).
Cf. A004186, A328447: largest & smallest representative of the class of n.

maxTerm = 103; (*maxTerm is the greatest term you wish to see*) permutationSet[n_Integer] := FromDigits /@ Permutations[IntegerDigits[n]]; permutationCongruentQ[x_Integer, y_Integer] := Sort[permutationSet[x]] == Sort[permutationSet[y]]; DeleteDuplicates[Range[maxTerm], permutationCongruentQ]
f[n_] := Block[{a = {0}, b = {DigitCount[0]}, i, w}, Do[w = DigitCount@ i; AppendTo[b, w]; If[! MemberQ[Most@ b, w], AppendTo[a, i]], {i, n}]; Rest@ a]; f@ 103 (* or faster: *)
Select[Range@ 103, LessEqual @@ IntegerDigits@ # || And[Take[IntegerDigits@ #, Last@ DigitCount@ # + 1] == Reverse@ Take[Sort@ IntegerDigits@ #, Last@ DigitCount@ # + 1], LessEqual @@ DeleteCases[IntegerDigits@ #, d_ /; d == 0]] &] (* Michael De Vlieger, Jul 14 2015 *)

is(n) = {my(d=digits(n),i); for(i=2,#d, if(d[i]!=0, d=vecextract(d,concat([1],vector(#d-i+1,j,i-1+j))); break));d==vecsort(d)||n/10^valuation(n,10)<10}
\\given an element n, in base b, find the next element from the sequence.
nxt(n,{b=10}) = {my(d = digits(n)); i = #d; while(i>0&&d[i]==b-1,i--); if(i>1, if(d[i]>0, d[i]++, d[i]=d[1];);for(j=i+1,#d,d[j]=d[i]), if(i==1, d[i]++;for(j=2,#d,d[j]=0), return(10^(#d))));sum(j=1,#d,d[j]*10^(#d-j))} \\ David A. Corneth, Apr 23 2016

select( is_A179239(n)={n==A328447(n)}, [0..200]) \\ M. F. Hasler, Oct 18 2019

from itertools import count, chain, islice
from sympy.utilities.iterables import combinations_with_replacement
def A179239_gen(): # generator of terms
    return chain((0,),(int(a+''.join(b)) for l in count(1) for a in '123456789' for b in combinations_with_replacement('0'+''.join(str(d) for d in range(int(a),10)),l-1)))
A179239_list = list(islice(A179239_gen(),31)) # Chai Wah Wu, Sep 13 2022

Prefixed with a(0) = 0 by M. F. Hasler, Oct 18 2019

A009996 Numbers with digits in nonincreasing order.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 20, 21, 22, 30, 31, 32, 33, 40, 41, 42, 43, 44, 50, 51, 52, 53, 54, 55, 60, 61, 62, 63, 64, 65, 66, 70, 71, 72, 73, 74, 75, 76, 77, 80, 81, 82, 83, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 110, 111, 200, 210, 211

Offset: 1

N. J. A. Sloane

Base-10 representation Sum_{i=0..m} d(i)*10^i has d(m) >= d(m-1) >= ... >= d(1) >= d(0).
These numbers might be called "Nialpdromes".
A004186(a(n)) = a(n). - Reinhard Zumkeller, Oct 31 2007

			As 10000 = C(10+6,10) - 6 + C(7+6,1+6) + C(5+5,1+5) + C(4+4,1+4) + C(3+3,1+3) + C(1+2,1+2) + C(0+1,1+1), C(0+0,1+0), a(10000) = 7543100.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
D. Applegate, M. LeBrun, and N. J. A. Sloane, Dismal Arithmetic, J. Int. Seq. 14 (2011) # 11.9.8.
David A. Corneth, Table of n, a(n) for n = 1..20000, Jun 03 2014
Eric Weisstein's World of Mathematics, Digit

Differs from A032873 and A032907.

Cf. A064222, A152054.

Select[Range[0,211], GreaterEqual@@IntegerDigits[#]&] (* Ray Chandler, Oct 25 2011 *)

is(n)=my(d=digits(n)); for(i=2,#d,if(d[i]>d[i-1],return(0))); 1 \\ Charles R Greathouse IV, Jan 02 2014

\\ This program is optimized for fast calculation of a(n) for large n.
a(n)={my(q,m=10,i,r=0);n--;while(binomial(m+1,10)<=n+m-9,m++);n-=binomial(m,10);n+=m-9;q=m-9;i=q;while(n>0,m=i;while(binomial(m+1,i)<=n,m++);r=10*r+m+1-i;n-=binomial(m,i);i--;);z=q-#digits(r);r*=10^z;r} \\ David A. Corneth, Jun 01 2014

\\recursive--feed an element a(n)>0 and it gives a(n+1).
nxt(n)={my(r,d=digits(n),y,t); if(d[#d]!=9,y=1; while(y-#d-1&&d[y]==9,y++); t=#d; forstep(i=t,y+1,-1,if(d[i-1]!=d[i],t=i-1;break)); if(t!=#d,d[t+1]++; for(i=t+2,#d,d[i]=0),d[y]++; for(i=y+1,#d,d[i]=0));r=d ,d=vector(#d+1); d[1]=1;for(i=2,#d,d[i]=0); r=d); sum(i=1,#r,10^(#r-i)*r[i])} \\ David A. Corneth, Jun 01 2014

from itertools import count, islice, combinations_with_replacement as mc
def agen(): # generator of terms
    yield 0
    for d in count(1):
        ni = (int("".join(m)) for m in mc("9876543210", d) if m[0]!="0")
        yield from sorted(ni)
print(list(islice(agen(), 70))) # Michael S. Branicky, Jun 24 2022

Binomial(n+k,k) = (n+k)!/(n!*k!). d(i) is the i-th digit of a(n). q is the number of digits of a(n). Find the highest m such that C(10 + m, 10) - m + 1 <= n. a(n) has m+1 digits. Set n = n - C(10+m,10). Find the highest d(m+1), then d(m), then ..., then d(1) each iteration such that C(d(m+1)+m+1,1+m+1)<=n. Then set n = n-C(d(m+1)+m+1,m+2). If n = 0 then stop. All remaining digits are 0.

Corrected by Rick L. Shepherd, Jun 06 2002

cp's OEIS Frontend

A352544 a(n) = n/2 if n is even, otherwise n + A004186(n), where A004186 = arrange digits in decreasing order.

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A096091 Numbers n with property that largest number formed from digits of n (A004186(n)) is divisible by smallest number formed from digits of n (A004185(n)).

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A283001 a(n) = (A004186(n) - n)/9.

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A337171 a(n) = A004186(n) mod n.

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A004086 Read n backwards (referred to as R(n) in many sequences).

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A151949 a(n) = image of n under the Kaprekar map n -> (n with digits sorted into descending order) - (n with digits sorted into ascending order).

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A004185 Arrange digits of n in increasing order, then (for n > 0) omit the zeros.

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