A009994 -id:A009994 - cp's OEIS frontend

A010785 Repdigit numbers, or numbers whose digits are all equal.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99, 111, 222, 333, 444, 555, 666, 777, 888, 999, 1111, 2222, 3333, 4444, 5555, 6666, 7777, 8888, 9999, 11111, 22222, 33333, 44444, 55555, 66666, 77777, 88888, 99999, 111111, 222222, 333333, 444444, 555555, 666666

Offset: 0

N. J. A. Sloane

Complement of A139819. - David Wasserman, May 21 2008
Subsequence of A134336 and of A178403. - Reinhard Zumkeller, May 27 2010
Subsequence of A193460. - Reinhard Zumkeller, Jul 26 2011
Intersection of A009994 and A009996. - David F. Marrs, Sep 29 2018
Beiler (1964) called these numbers "monodigit numbers". The term "repdigit numbers" was used by Trigg (1974). - Amiram Eldar, Jan 21 2022

Albert H. Beiler, Recreations in the Theory of Numbers, Dover, New York, 1964, p. 83.

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Eric F. Bravo, Carlos A. Gómez and Florian Luca, Product of Consecutive Tribonacci Numbers With Only One Distinct Digit, J. Int. Seq., Vol. 22 (2019), Article 19.6.3.
Eric Fernando Bravo, On concatenations of Padovan and Perrin numbers, Math. Commun. (2023) Vol 28, 105-119.
Mahadi Ddamulira, Repdigits as sums of three balancing numbers, Mathematica Slovaca, (2019), hal-02405969.
Mahadi Ddamulira, Padovan numbers that are concatenations of two distinct repdigits, arXiv:2003.10705 [math.NT], 2020.
Mahadi Ddamulira, Tribonacci numbers that are concatenations of two repdigits, hal-02547159, Mathematics [math] / Number Theory [math.NT], 2020.
Mahadi Ddamulira, Padovan numbers that are concatenations of two distinct repdigits, Mathematica Slovaca, Vol. 71, No. 2 (2021), pp. 275-284.
Bart Goddard and Jeremy Rouse, Sum of two repdigits a square, arXiv:1607.06681 [math.NT], 2016. Mentions this sequence.
Bir Kafle, Florian Luca and Alain Togbé, Triangular Repblocks, Fibonacci Quart., Vol. 56, No. 4 (2018), pp. 325-328.
Bir Kafle, Florian Luca and Alain Togbé, Pentagonal and heptagonal repdigits, Annales Mathematicae et Informaticae, Vol. 52 (2020), pp. 137-145.
Benedict Vasco Normenyo, Bir Kafle, and Alain Togbé, Repdigits as Sums of Two Fibonacci Numbers and Two Lucas Numbers, Integers, Vol. 19 (2019), Article A55.
Salah Eddine Rihane and Alain Togbé, Repdigits as products of consecutive Padovan or Perrin numbers, Arab. J. Math., Vol. 10 (2021), pp. 469-480.
Charles W. Trigg, Infinite sequences of palindromic triangular numbers, The Fibonacci Quarterly, Vol. 12, No. 2 (1974), pp. 209-212.
Eric Weisstein's World of Mathematics, Repdigit.
Wikipedia, Repdigit.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,-10).

Cf. A000005, A009994, A009996, A010879, A037904, A047842, A059995, A065444, A134336, A139819, A151949, A178401, A178403, A180410, A193459, A193460, A202022, A227362, A244112.

a010785 n = a010785_list !! n
a010785_list = 0 : r [1..9] where
   r (x:xs) = x : r (xs ++ [10*x + x `mod` 10])
-- Reinhard Zumkeller, Jul 26 2011

[(n-9*Floor((n-1)/9))*(10^Floor((n+8)/9)-1)/9: n in [0..50]]; // Vincenzo Librandi, Nov 10 2014

A010785 := proc(n)
    (n-9*floor(((n-1)/9)))*((10^(floor(((n+8)/9)))-1)/9) ;
end proc:
seq(A010785(n), n = 0 .. 100); # Robert Israel, Nov 09 2014

fQ[n_]:=Module[{id=IntegerDigits[n]}, Length[Union[id]]==1]; Select[Range[0,10000], fQ] (* Vladimir Joseph Stephan Orlovsky, Dec 29 2010 *)
Union[FromDigits/@Flatten[Table[PadRight[{},i,n],{n,0,9},{i,6}],1]] (* or *) LinearRecurrence[{0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,-10}, {0,1,2,3,4,5,6,7,8,9,11,22,33,44,55,66,77,88},40] (* Harvey P. Dale, Dec 28 2011 *)
Union@ Flatten@ Table[k (10^n - 1)/9, {k, 0, 9}, {n, 6}] (* Robert G. Wilson v, Oct 09 2014 *)
Table[(n - 9 Floor[(n-1)/9]) (10^Floor[(n+8)/9] - 1)/9, {n, 0, 50}] (* José de Jesús Camacho Medina, Nov 06 2014 *)

a(n)=10^((n+8)\9)\9*((n-1)%9+1) \\ Charles R Greathouse IV, Jun 15 2011

nxt(n,t=n%10)=if(t<9,n*(t+1),n*10+9)\t \\ Yields the term a(k+1) following a given term a(k)=n. M. F. Hasler, Jun 24 2016

is(n)={1==#Set(digits(n))}
inv(n) = 9*#Str(n) + n%10 - 9 \\ David A. Corneth, Jun 24 2016

def a(n): return 0 if n == 0 else int(str((n-1)%9+1)*((n-1)//9+1))
print([a(n) for n in range(55)]) # Michael S. Branicky, Dec 29 2021

print([0]+[int(d*r) for r in range(1, 7) for d in "123456789"]) # Michael S. Branicky, Dec 29 2021

# without string operations
def a(n): return 0 if n == 0 else (10**((n-1)//9+1)-1)//9*((n-1)%9+1)
print([a(n) for n in range(55)]) # Michael S. Branicky, Nov 03 2023

A037904(a(n)) = 0. - Reinhard Zumkeller, Dec 14 2007
A178401(a(n)) > 0. - Reinhard Zumkeller, May 27 2010
From Reinhard Zumkeller, Jul 26 2011: (Start)
For n > 0: A193459(a(n)) = A000005(a(n)).
for n > 10: a(n) mod 10 = floor(a(n)/10) mod 10.
A010879(n) = A010879(A059995(n)). (End)
A202022(a(n)) = 1. - Reinhard Zumkeller, Dec 09 2011
a(0)=0, a(1)=1, a(2)=2, a(3)=3, a(4)=4, a(5)=5, a(6)=6, a(7)=7, a(8)=8, a(9)=9, a(10)=11, a(11)=22, a(12)=33, a(13)=44, a(14)=55, a(15)=66, a(16)=77, a(17)=88, a(n) = 11*a(n-9) - 10*a(n-18). - Harvey P. Dale, Dec 28 2011
A151949(a(n)) = 0; A180410(a(n)) = A227362(a(n)). - Reinhard Zumkeller, Jul 09 2013
a(n) = (n - 9*floor((n-1)/9))*(10^floor((n+8)/9) - 1)/9. - José de Jesús Camacho Medina, Nov 06 2014
G.f.: x*(1+2*x+3*x^2+4*x^3+5*x^4+6*x^5+7*x^6+8*x^7+9*x^8)/((1-x^9)*(1-10*x^9)). - Robert Israel, Nov 09 2014
A047842(a(n)) = A244112(a(n)). - Reinhard Zumkeller, Nov 11 2014
Sum_{n>=1} 1/a(n) = (7129/2520) * A065444 = 3.11446261209177581335... - Amiram Eldar, Jan 21 2022

Name clarified by Jon E. Schoenfield, Nov 10 2023

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

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

A036839 RATS(n): Reverse Add Then Sort the digits.

Original entry on oeis.org

0, 2, 4, 6, 8, 1, 12, 14, 16, 18, 11, 22, 33, 44, 55, 66, 77, 88, 99, 11, 22, 33, 44, 55, 66, 77, 88, 99, 11, 112, 33, 44, 55, 66, 77, 88, 99, 11, 112, 123, 44, 55, 66, 77, 88, 99, 11, 112, 123, 134, 55, 66, 77, 88, 99, 11, 112, 123, 134, 145, 66, 77

Offset: 0

N. J. A. Sloane, Jan 19 2002

a(n) = RATS(n), not RATS(a(n-1)).

Row 10 of A288535. - Andrey Zabolotskiy, Jun 14 2017

			1 -> 1 + 1 = 2, so a(1) = 2; 3 -> 3 + 3 = 6, so a(3) = 6.

Indranil Ghosh, Table of n, a(n) for n = 0..50000 (terms 0..1000 from T. D. Noe)
R. K. Guy, Conway's RATS and other reversals, Unsolved Problems Column, American Math. Monthly, Vol. 96, pp. 425-428, May 1989.
Eric Weisstein's World of Mathematics, RATS Sequence

Cf. A004000, A004185, A288535.

Cf. A161593, A114611, A009994, A004086.

a036839 = a004185 . a056964  -- Reinhard Zumkeller, Mar 14 2012

read transforms; RATS := n -> digsort(n + digrev(n));

FromDigits[Sort[IntegerDigits[#+FromDigits[Reverse [IntegerDigits[#]]]]]] & /@Range[0,80]  (* Harvey P. Dale, Mar 26 2011 *)

def A036839(n):
    x = str(n+int(str(n)[::-1]))
    return int("".join(sorted(x))) # Indranil Ghosh, Jan 28 2017

Form m by Reversing the digits of n, Add m to n Then Sort the digits of the sum into increasing order to get a(n).

a(n) = A004185(A056964(n)). [Reinhard Zumkeller, Mar 14 2012]

A028864 Primes with digits in nondecreasing order.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 37, 47, 59, 67, 79, 89, 113, 127, 137, 139, 149, 157, 167, 179, 199, 223, 227, 229, 233, 239, 257, 269, 277, 337, 347, 349, 359, 367, 379, 389, 449, 457, 467, 479, 499, 557, 569, 577, 599, 677, 1117, 1123, 1129

Offset: 1

Patrick De Geest

Identical digits are acceptable, e.g., 1117 is in the sequence. - Harvey P. Dale, Aug 16 2011

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (terms n = 1..1000 from T. D. Noe)

Cf. A009994, A052015, A028867, A052014, A028905.

[p:p in PrimesUpTo(1200)| Reverse(Intseq(p)) eq Sort(Intseq(p))]; // Marius A. Burtea, Nov 29 2019

daoQ[n_] := Count[Differences[IntegerDigits[n]], ?(# < 0 &)] == 0; Select[Prime[Range[200]], daoQ] (* _Harvey P. Dale, Aug 16 2011 *)
Select[Prime[Range[200]],Min[Differences[IntegerDigits[#]]]>-1&] (* Harvey P. Dale, Mar 02 2023 *)

select(n->n=digits(n); n==vecsort(n), primes(500)) \\ Charles R Greathouse IV, Mar 15 2014

from itertools import count, islice, combinations_with_replacement
from sympy import isprime
def A028864_gen(): # generator of terms
    yield from (2,3,5,7)
    a, b = {'1':0,'2':1,'3':1,'4':2,'5':2,'6':2,'7':2,'8':3,'9':3}, (1,3,7,9)
    for l in count(1):
        for d in combinations_with_replacement('123456789',l):
            k = 10*int(''.join(d))
            for e in b[a[d[-1]]:]:
                if isprime(m:=k+e):
                    yield m
A028864_list = list(islice(A028864_gen(),30)) # Chai Wah Wu, Dec 25 2023

j=2; y=as.bigz(c()); while(j<1000) {
x=sort(as.numeric(strsplit(as.character(j),spl="")[[1]]),decr=F)
if(j==paste(x[x>0],collapse="")) y=c(y,j)
j=nextprime(j)
} //  Christian N. K. Anderson, Apr 04 2013

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

prime(n) = A028905(n) if prime(n) is in this sequence. - Alonso del Arte, Nov 25 2019

Definition corrected by Omar E. Pol, Mar 22 2012

A179255 Number of partitions of n into distinct parts such that the successive differences of consecutive parts are nondecreasing.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 5, 5, 8, 9, 10, 13, 15, 16, 22, 24, 26, 33, 36, 39, 50, 54, 58, 70, 77, 83, 100, 109, 116, 137, 150, 159, 186, 202, 216, 249, 270, 288, 328, 355, 379, 428, 462, 491, 554, 597, 633, 707, 760, 807, 899, 964, 1020, 1127, 1211, 1282, 1412, 1512, 1596, 1750, 1873, 1976, 2160, 2305, 2434, 2652, 2826, 2978

Offset: 0

Joerg Arndt, Jan 05 2011

nonn

Partitions into distinct parts (p(1), p(2), ..., p(m)) such that p(k-1) - p(k-2) <= p(k) - p(k-1) for all k >= 3.

			There are a(17) = 26 such partitions of 17:
01:  [ 1 2 3 4 7 ]
02:  [ 1 2 3 11 ]
03:  [ 1 2 4 10 ]  *
04:  [ 1 2 5 9 ]   *
05:  [ 1 2 14 ]   *
06:  [ 1 3 5 8 ]
07:  [ 1 3 13 ]   *
08:  [ 1 4 12 ]   *
09:  [ 1 5 11 ]   *
10:  [ 1 16 ]   *
11:  [ 2 3 4 8 ]
12:  [ 2 3 5 7 ]
13:  [ 2 3 12 ]   *
14:  [ 2 4 11 ]   *
15:  [ 2 5 10 ]   *
16:  [ 2 15 ]   *
17:  [ 3 4 10 ]   *
18:  [ 3 5 9 ]   *
19:  [ 3 14 ]   *
20:  [ 4 5 8 ]   *
21:  [ 4 13 ]   *
22:  [ 5 12 ]   *
23:  [ 6 11 ]   *
24:  [ 7 10 ]   *
25:  [ 8 9 ]   *
26:  [ 17 ]   *
The 21 partitions marked with * have strictly increasing differences, see the example for A179254.
- _Joerg Arndt_, Mar 31 2014

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..1000 (terms 0..241 from Joerg Arndt)

Cf. A009994.
Cf. A179254 (strictly increasing differences), A179269, A007294.
Cf. A240026 (partitions with nondecreasing differences), A240027 (partitions with strictly increasing differences), A320382.

def partition(n, min, max)
  return [[]] if n == 0
  [max, n].min.downto(min).flat_map{|i| partition(n - i, min, i - 1).map{|rest| [i, *rest]}}
end
def f(n)
  return 1 if n == 0
  cnt = 0
  partition(n, 1, n).each{|ary|
    ary0 = (1..ary.size - 1).map{|i| ary[i - 1] - ary[i]}
    cnt += 1 if ary0.sort == ary0.reverse
  }
  cnt
end
def A179255(n)
  (0..n).map{|i| f(i)}
end
p A179255(50) # Seiichi Manyama, Oct 12 2018

def A179255(n):
    has_nondecreasing_diffs = lambda x: min(differences(x,2)) >= 0
    allowed = lambda x: len(x) < 3 or has_nondecreasing_diffs(x)
    return len([x for x in Partitions(n,max_slope=-1) if allowed(x[::-1])])
# D. S. McNeil, Jan 06 2011

A108662 Numbers whose sum of squares of digits is a prime.

Original entry on oeis.org

11, 12, 14, 16, 21, 23, 25, 27, 32, 38, 41, 45, 49, 52, 54, 56, 58, 61, 65, 72, 78, 83, 85, 87, 94, 101, 102, 104, 106, 110, 111, 113, 119, 120, 126, 131, 133, 137, 140, 146, 159, 160, 162, 164, 166, 168, 173, 179, 186, 191, 195, 197, 199, 201, 203, 205, 207, 210

Offset: 1

Zak Seidov, Jun 16 2005

If m is in the sequence, then so are 10*m and any anagram (even with adding zeros between digits) of m. E.g., 12 is a term, hence 21, 102, 120, 201, 10020 all are here.

A sequence of primitive terms is of interest. It starts with 11, 12, 14, 16, 23, 25, 27, 38, 45, 49, 56, 58, 78, 111, 113, 119, 126, 133, 137, 146, 159, 166, 168, 179, 199. Note that digits are in nondecreasing order. - Zak Seidov, Dec 31 2013

			23 is in the sequence because 2^2 + 3^2 = 13 is a prime.

Harvey Dale and Zak Seidov, Table of n, a(n) for n = 1..10000 [First 1000 terms from Harvey Dale]

Cf. A003132, A009994, A052034, A234021.

Select[Range[300],PrimeQ[Total[IntegerDigits[#]^2]]&] (* Harvey P. Dale, May 25 2012 *)

isok(n) = isprime(norml2(digits(n))); \\ Michel Marcus, Jan 09 2019

A152054 Bouncy numbers (numbers whose digits are in neither increasing nor decreasing order).

Original entry on oeis.org

101, 102, 103, 104, 105, 106, 107, 108, 109, 120, 121, 130, 131, 132, 140, 141, 142, 143, 150, 151, 152, 153, 154, 160, 161, 162, 163, 164, 165, 170, 171, 172, 173, 174, 175, 176, 180, 181, 182, 183, 184, 185, 186, 187, 190, 191, 192, 193, 194, 195, 196

Offset: 1

Jerome Abela (Jerome.Abela(AT)gmail.com), Nov 22 2008

Complement of union of A009994 and A009996. - Ray Chandler, Oct 25 2011

Number of n-digit bouncy numbers is 9*10^(n-1) - (n+18)*binomial(n+8, 8)/9 + 10. - Altug Alkan, Oct 02 2018

David F. Marrs, Table of n, a(n) for n = 1..10000
Project Euler, Non-bouncy numbers Problem 113

Cf. A152464. - Jon E. Schoenfield, Dec 06 2008

Cf. A009994, A009996, A204692.

Select[Range[0,200], !LessEqual@@IntegerDigits[#] && !GreaterEqual@@IntegerDigits[#]&] (* Ray Chandler, Oct 25 2011 *)
bnQ[n_]:=Module[{didn=Differences[IntegerDigits[n]]},Count[didn,?(#>0&)]>0 && Count[didn,?(#<0&)]>0]; Select[Range[100,200],bnQ] (* Harvey P. Dale, Jun 13 2020 *)

a = 1
b = 100
while a != 51:
    if str(b) != ''.join(sorted(str(b))) and str(b) != ''.join(sorted(str(b)))[::-1]:
        print(b)
        a += 1
    b += 1
# David F. Marrs, Sep 25 2018

from itertools import count, islice
def A152054_gen(startvalue=1): # generator of terms >= startvalue
    for n in count(max(startvalue,1)):
        l = len(s:=tuple(int(d) for d in str(n)))
        for i in range(1,l-1):
            if (s[i-1]-s[i])*(s[i]-s[i+1]) < 0:
                yield n
                break
A152054_list = list(islice(A152054_gen(),30)) # Chai Wah Wu, Jul 28 2023

More terms from Jon E. Schoenfield, Dec 06 2008

A239016 Numbers not larger than any rotation of their digits.

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, 123, 124, 125, 126, 127, 128, 129, 132

Offset: 1

M. F. Hasler, Mar 08 2014

The numbers with nonincreasing digits, A009994, form a subsequence which first differs at a(73)=132 (not in A009994) from this one.
This sequence is a subsequence of A072544: numbers whose smallest decimal digit is also the initial digit. A072544(65)=121 is the first such number not in this sequence.
This criterion involving "rotation" is part of the characterization of Lyndon words, see e.g. A102659, A102660, A210584, A210585. All of these are subsequences of this sequence. For example, A102659 = A213969 intersect A239016.

			The number 10 is excluded from this sequence because its "rotation" 01 is smaller than the number itself.
The same is the case for any number whose first digit is not the smallest one: rotating a smaller digit to the front will always yield a smaller number, independently of the other digits. For this reason, all terms must be in A072544.
a(73)=132 is in the sequence because the nontrivial rotations of its digits are 321 and 213, both larger than 132.

is_A239016(n)=vecsort(d=digits(n))==d||!for(i=1,#d-1,n>[1,10^(#d-i)]*divrem(n,10^i)&&return)

def ok(n):
    s = str(n)
    if "".join(sorted(s)) == s: return True
    return all(n <= int(s[i:] + s[:i]) for i in range(1, len(s)))
print(list(filter(ok, range(133)))) # Michael S. Branicky, Aug 21 2021

A261370 Permutation of nonnegative integers where a number having digits in nondescending order is followed by all numbers having the same digits arranged in increasing order.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 21, 13, 31, 14, 41, 15, 51, 16, 61, 17, 71, 18, 81, 19, 91, 20, 22, 23, 32, 24, 42, 25, 52, 26, 62, 27, 72, 28, 82, 29, 92, 30, 33, 34, 43, 35, 53, 36, 63, 37, 73, 38, 83, 39, 93, 40, 44, 45, 54, 46, 64, 47, 74, 48, 84

Offset: 0

David A. Corneth, Aug 17 2015

If a number contains a zero, then some permutation will yield a number with a leading zero, which is already in the sequence without the leading zero. So that permutation is not included. For example, 102 contains a zero, so 012 and 021 are permutations of these numbers' digits. But they are actually 12 and 21, which are already in the sequence. This leaves 120, 201 and 210 to be added to the sequence after 102.
From Rémy Sigrist, May 01 2017 : (Start)
- This sequence is to base 10 what A187769 is to base 2,
- Beyond the initial 0, this sequence can be seen as an irregular table, where the n-th row corresponds to the permutation class of A179239(n).
(End)

David A. Corneth, Table of n, a(n) for n = 0..10000

Cf. A009994, A179239, A187769.

a = {0}; f[n_] := Block[{w = Sort@ Permutations@ IntegerDigits@ n}, w = Delete[w, Position[First /@ w, 0]]]; Do[If[! MemberQ[a, n], AppendTo[a, FromDigits /@ f@ n]], {n, 105}]; DeleteDuplicates@ Flatten@ a (* Michael De Vlieger, Sep 07 2015 *)

cp's OEIS Frontend

A010785 Repdigit numbers, or numbers whose digits are all equal.

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

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

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A036839 RATS(n): Reverse Add Then Sort the digits.

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A028864 Primes with digits in nondecreasing order.

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