A193581 -id:A193581 - cp's OEIS frontend

A193582 Persistence of n for sort-and-subtract (A193581).

0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 3, 2, 2, 2, 2, 2, 1, 1, 1, 1, 3, 3, 2, 2, 2, 2, 2, 1, 1, 1, 3, 3, 3, 2, 2, 2

Offset: 0

Reinhard Zumkeller, Aug 10 2011

Number of sort-and-subtract steps to reach 0 when starting with n.

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

Cf. A004185.

a193582 n = length $ takeWhile (> 0) $ iterate a193581 n
a193582_list = map a193582 [0..]

A009994 Numbers with digits in nondecreasing order.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 19, 22, 23, 24, 25, 26, 27, 28, 29, 33, 34, 35, 36, 37, 38, 39, 44, 45, 46, 47, 48, 49, 55, 56, 57, 58, 59, 66, 67, 68, 69, 77, 78, 79, 88, 89, 99, 111, 112, 113, 114, 115, 116, 117, 118, 119, 122

Offset: 1

N. J. A. Sloane

Record values and occurrences of A004185. - Reinhard Zumkeller, Dec 05 2009
A193581(a(n)) = 0. - Reinhard Zumkeller, Aug 10 2011
This sequence was used by the U.S. Bureau of the Census in the mid-1950s to numerically code the alphabetical list of counties within a state (with some modifications for Texas). The 3-digit code has a "self-policing element" built into it and "was fairly effective in detecting the transposition of 2 digits." (Hanna 1959). - Randy A. Becker, Dec 11 2017

Amarnath Murthy and Robert J. Clarke, Some Properties of Staircase sequence, Mathematics & Informatics Quarterly, Volume 11, No. 4, November 2001.
Frank A. Hanna, The Compilation of Manufacturing Statistics. U.S. Department of Commerce, Bureau of the Census, 1959.

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Reinhard Zumkeller)
David Applegate, Marc LeBrun, N. J. A. Sloane, Dismal Arithmetic, J. Int. Seq. 14 (2011) # 11.9.8.
David Radcliffe and Brendan McKay, Re: A009994, SeqFan list, Jul 29 2019
Eric Weisstein's World of Mathematics, Digit.
Index entries for 10-automatic sequences.

Apart from the first term, a subsequence of A052382. A254143 is a subsequence.

Cf. A152054, A036839.

import Data.Set (fromList, deleteFindMin, insert)
a009994 n = a009994_list !! n
a009994_list = 0 : f (fromList [1..9]) where
   f s = m : f (foldl (flip insert) s' $ map (10*m +) [m `mod` 10 ..9])
         where (m,s') = deleteFindMin s
-- Reinhard Zumkeller, Aug 10 2011

[k:k in [0..122]|Sort(Intseq(k)) eq Reverse(Intseq(k))]; // Marius A. Burtea, Jul 28 2019

A[0]:= [0]: A[1]:= [$1..9]:
for d from 2 to 4 do
  A[d]:= map(t -> seq(10*t+i,i=(t mod 10) .. 9), A[d-1]):
od:
seq(op(A[d]),d=0..4); # Robert Israel, Jul 28 2019

Select[Range[0, 125], LessEqual@@IntegerDigits[#] &] (* Ray Chandler, Oct 25 2011 *)

is(n)=n=digits(n);n==vecsort(n) \\ Charles R Greathouse IV, Dec 03 2013

from itertools import combinations_with_replacement
def A009994generator():
    yield 0
    l = 1
    while True:
        for i in combinations_with_replacement('123456789',l):
            yield int(''.join(i))
        l += 1 # Chai Wah Wu, Nov 11 2015

def hasDigitsSorted(n: Int): Boolean = {
  val digSort = Integer.parseInt(n.toString.toCharArray.sorted.mkString)
  n == digSort
}
(0 to 200).filter(hasDigitsSorted()) // _Alonso del Arte, Apr 20 2020

a(n) >> exp(n^(1/10)). - Charles R Greathouse IV, Mar 15 2014

a(n) ~ 10^((9! n)^(1/9) - 5), since 10^(d - 1) <= a(n) < 10^d for binomial(d + 8, 9) < n <= binomial(d + 9, 9) = (d + 5 - epsilon)^9 / 9!. Using epsilon = 10/(3n) + o(1/n) gives more precise estimate. [Following Radcliffe and McKay, cf. SeqFan list.] - M. F. Hasler, Jul 30 2019

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

A009995 Numbers with digits in strictly decreasing order. From the Macaulay expansion of n.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 20, 21, 30, 31, 32, 40, 41, 42, 43, 50, 51, 52, 53, 54, 60, 61, 62, 63, 64, 65, 70, 71, 72, 73, 74, 75, 76, 80, 81, 82, 83, 84, 85, 86, 87, 90, 91, 92, 93, 94, 95, 96, 97, 98, 210, 310, 320, 321, 410, 420, 421, 430, 431, 432, 510, 520, 521, 530

Offset: 1

N. J. A. Sloane

There are precisely 1023 terms (corresponding to every nonempty subset of {0..9}).
A178788(a(n)) = 1. - Reinhard Zumkeller, Jun 30 2010
A193581(a(n)) > 0 for n > 9. - Reinhard Zumkeller, Aug 10 2011
A227362(a(n)) = a(n). - Reinhard Zumkeller, Jul 09 2013
For a fixed natural number r, any natural number n has a unique "Macaulay expansion" n = C(a_r,r)+C(a_{r-1},r-1)+...+C(a_1,1) with a_r > a_{r-1} > ... > a_1 >= 0. If r=10, concatenating the digits a_r, ..., a_1 gives the present sequence. The representation is valid for all n, but the concatenation only makes sense if all the a_i are < 10. - N. J. A. Sloane, Apr 05 2014
a(n) = A262557(A263327(n)); a(A263328(n)) = A262557(n). - Reinhard Zumkeller, Oct 15 2015

Reinhard Zumkeller, Table of n, a(n) for n = 1..1023
B. Sury, Macaulay Expansion, Amer. Math. Monthly 121 (2014), no. 4, 359--360. MR3183022. [See p. 359. - _N. J. A. Sloane_, Apr 05 2014]
Eric Weisstein's World of Mathematics, Digit.

Cf. A009993.

Cf. A262557 (sorted lexicographically), A263327, A263328.

import Data.Set (fromList, minView, insert)
a009995 n = a009995_list !! n
a009995_list = 0 : f (fromList [1..9]) where
   f s = case minView s of
         Nothing     -> []
         Just (m,s') -> m : f (foldl (flip insert) s' $
                              map (10*m +) [0..m `mod` 10 - 1])
-- Reinhard Zumkeller, Aug 10 2011

Sort@ Flatten@ Table[FromDigits /@ Subsets[ Range[9, 0, -1], {n}], {n, 10}] (* Zak Seidov, May 10 2006 *)

is(n)=fromdigits(vecsort(digits(n),,12))==n \\ Charles R Greathouse IV, Apr 16 2015

A108782 Difference between n and the largest number with the same digit set as n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 9, 18, 27, 36, 45, 54, 63, 72, 0, 0, 0, 9, 18, 27, 36, 45, 54, 63, 0, 0, 0, 0, 9, 18, 27, 36, 45, 54, 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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 9, 0, 0

Offset: 0

Zak Seidov, Jun 29 2005

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A004186, A151949, A193581.

FromDigits[Sort[IntegerDigits[n], Greater]] - n (Rob Pratt)
Table[Max[FromDigits/@ Permutations[IntegerDigits[n]]]-n, {n, 150}]

a(n) = A004186(n) - n. - Seiichi Manyama, Sep 25 2018

a(0)=0 prepended by Seiichi Manyama, Sep 25 2018

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A193582 Persistence of n for sort-and-subtract (A193581).

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A009994 Numbers with digits in nondecreasing order.

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

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A009995 Numbers with digits in strictly decreasing order. From the Macaulay expansion of n.

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A108782 Difference between n and the largest number with the same digit set as n.

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