A009994 -id:A009994 - cp's OEIS frontend

A329299 Numbers whose digits are in nondecreasing order in bases 9 and 10.

0, 1, 2, 3, 4, 5, 6, 7, 8, 11, 12, 13, 14, 15, 16, 17, 22, 23, 24, 25, 26, 33, 34, 35, 44, 111, 112, 113, 114, 115, 116, 122, 123, 124, 125, 133, 134, 188, 222, 223, 224, 233, 277, 278, 366, 367, 368, 377, 455, 456, 457, 458, 466, 467, 556, 557, 566, 1113

Offset: 1

Jon E. Schoenfield, Nov 17 2019

a(91) = 12555566 is the largest term < 10^10000 (which is a 10480-digit number in base 9). But can it be proved that 12555566 is the final term of the sequence?

			Sequence includes, respectively, 9, 16, 32, and 11 terms that are 1-, 2-, 3-, and 4- digit terms in both bases, and the following:
  a(69) =    14777 =    22238_9
  a(70) =    15677 =    23448_9
  a(71) =    22234 =    33444_9
  a(72) =    22235 =    33445_9
  a(73) =    22236 =    33446_9
  a(74) =    22237 =    33447_9
  a(75) =    22238 =    33448_9
  a(76) =    22244 =    33455_9
  a(77) =    22245 =    33456_9
  a(78) =    22246 =    33457_9
  a(79) =    22247 =    33458_9
  a(80) =    22255 =    33467_9
  a(81) =    22256 =    33468_9
  a(82) =    22335 =    33566_9
  a(83) =    22336 =    33567_9
  a(84) =    22337 =    33568_9
  a(85) =    22345 =    33577_9
  a(86) =    22346 =    33578_9
  a(87) =    22355 =    33588_9
  a(88) =    44468 =    66888_9
  a(89) =   222344 =   367888_9
  a(90) =  1233467 =  2278888_9
  a(91) = 12555566 = 25555888_9

Intersection of A023751 (base 9) and A009994 (base 10). Numbers whose digits are in nondecreasing order in bases b and b+1: A329294 (b=4), A329295 (b=5), A329296 (b=6), A329297 (b=7), A329299 (b=8), this sequence (b=9). See A329300 for the (apparently) largest term of each of these sequences.

filter:= proc(n) local L;
  L:= convert(n,base,10);
  `and`(seq(L[i+1]<=L[i],i=1..nops(L)-1))
end proc:
ND[1]:= [$1..8]: R:= $0..8:
for d from 2 to 10 do
  ND[d]:= map(t -> seq(9*t+r, r=(t mod 9) ..8), ND[d-1]);
  R:= R, op(select(filter, ND[d]));
od:
R; # Robert Israel, Nov 20 2019

Select[Range[0,1200],Min[Differences[IntegerDigits[#]]]>-1&& Min[ Differences[ IntegerDigits[ #,9]]]>-1&] (* Harvey P. Dale, Oct 14 2022 *)

A028820 Squares with digits in nondecreasing order.

Original entry on oeis.org

0, 1, 4, 9, 16, 25, 36, 49, 144, 169, 225, 256, 289, 1156, 1225, 1369, 1444, 4489, 6889, 11236, 11449, 13456, 13689, 27889, 33489, 111556, 112225, 113569, 134689, 146689, 344569, 444889, 2666689, 2778889, 11115556, 11122225, 11135569

Offset: 1

Patrick De Geest

Number of terms less than 10^k, beginning with k=0: 1, 4, 8, 13, 19, 25, 32, 34, 42, 43, 50, 53, 61, 62, 71, 72, 82, 83, 94, 95, …, .

Like all squares the ending digits can be 0, 1, 4, 5, 6 or 9. Here is the tally of the list of terms < 10^19: {0, 1}, {1, 1}, {4, 4}, {5, 10}, {6, 13}, {9, 66}. Robert G. Wilson v, Jan 01 2014

Robert G. Wilson v and Chai Wah Wu, Table of n, a(n) for n = 1..428 (n = 1..106 from Robert G. Wilson v).
Patrick De Geest, Palindromic Squares in bases 2 to 17

Cf. A028819, A028821, A028822, A237424.

Intersection of A000290 and A009994.

Select[Range[0,4000]^2,Min[Differences[IntegerDigits[#]]]>-1&] (* Harvey P. Dale, Dec 31 2013 *)
Select[Range[0,10^4]^2,LessEqual@@IntegerDigits[#]&] (* Ray Chandler, Jan 06 2014 *)

mono(n)=n=eval(Vec(Str(n)));for(i=2,#n,if(n[i]Charles R Greathouse IV, Aug 22 2011

from itertools import combinations_with_replacement
from gmpy2 import is_square
A028820_list = [0] + [n for n in (int(''.join(i)) for l in range(1,11) for i in combinations_with_replacement('123456789',l)) if is_square(n)] # Chai Wah Wu, Dec 07 2015

a(n) = A028819(n)^2. - Ray Chandler, Jan 06 2014

Definition edited by Zak Seidov, Dec 31 2013

A193581 Sort-and-subtract: a(n) = n - A004185(n).

Original entry on oeis.org

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

Offset: 0

Reinhard Zumkeller, Aug 10 2011

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

Cf. A009994, A004185, A065641.

a193581 n = n - a004185 n
a193581_list = map a193581 [0..]

ss[n_]:=n-FromDigits[Sort[Select[IntegerDigits[n],#>0&]]]; Array[ss,80,0] (* Harvey P. Dale, Aug 13 2013 *)

a(A009994(n)) = 0.

For n > 10, a(A009995(n)) > 0.

A254143 Products of any two not necessarily distinct terms of A237424.

Original entry on oeis.org

1, 4, 7, 16, 28, 34, 37, 49, 67, 136, 148, 238, 259, 268, 334, 337, 367, 469, 667, 1156, 1258, 1336, 1348, 1369, 1468, 2278, 2338, 2359, 2479, 2569, 2668, 3334, 3337, 3367, 3667, 4489, 4669, 6667, 11356, 11458, 12358, 12469, 12478, 13336, 13348, 13468, 13579

Offset: 1

Reinhard Zumkeller, Jan 28 2015

nonn

Digits are in nondecreasing order for all terms in decimal representation;
a(396) = 1123456789 = 3367 * 333667 is the smallest term containing all nonzero decimal digits: A254323(396) = 123456789;
A254323(n) = A137564(a(n)).

			Initial terms of A237424: 1, 4, 7, 34, 37, 67, 334, 337, 367, 667, 3334 ...
.  n | a(n) = A237424(i) * A237424(j)
. ---+-------------------------------
.  1 |    1 = 1 * 1   = A237424(1)^2
.  2 |    4 = 1 * 4   = A237424(1) * A237424(2)
.  3 |    7 = 1 * 7   = A237424(1) * A237424(3)
.  4 |   16 = 4 * 4   = A237424(2)^2
.  5 |   28 = 4 * 7   = A237424(2) * A237424(3)
.  6 |   34 = 1 * 34  = A237424(1) * A237424(4)
.  7 |   37 = 4 * 37  = A237424(1) * A237424(5)
.  8 |   49 = 7 * 7   = A237424(3)^2
.  9 |   67 = 1 * 67  = A237424(1) * A237424(6)
. 10 |  136 = 4 * 34  = A237424(2) * A237424(4)
. 11 |  148 = 4 * 37  = A237424(2) * A237424(5)
. 12 |  238 = 7 * 34  = A237424(3) * A237424(4)
. 13 |  259 = 7 * 37  = A237424(3) * A237424(5)
. 14 |  268 = 4 * 67  = A237424(2) * A237424(6)
. 15 |  334 = 1 * 334 = A237424(1) * A237424(7)
. 16 |  337 = 1 * 337 = A237424(1) * A237424(8)
. 17 |  367 = 1 * 367 = A237424(1) * A237424(9)
. 18 |  469 = 7 * 67  = A237424(3) * A237424(6)
. 19 |  667 = 1 * 34  = A237424(1) * A237424(10)
. 20 | 1156 = 34 * 34 = A237424(4)^2
see link for more.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Reinhard Zumkeller, First 10000 products of any two terms of A237424

Subsequence of A009994.

Cf. A237424, A254323, A137564, A254338 (initial digits), A254339 (final digits).

import Data.Set (empty, fromList, deleteFindMin, union)
import qualified Data.Set as Set (null)
a254143 n = a254143_list !! (n-1)
a254143_list = f a237424_list [] empty where
   f xs'@(x:xs) zs s
     | Set.null s || x < y = f xs zs' (union s $ fromList $ map (* x) zs')
     | otherwise           = y : f xs' zs s'
     where zs' = x : zs
           (y, s') = deleteFindMin s

listA237424(lim)=my(v=List(),a,t); while(1, for(b=0,a, t=(10^a+10^b+1)/3; if(t>lim, return(Set(v))); listput(v, t)); a++)
list(lim)=my(v=List(),u=listA237424(lim),t); for(i=1,#u, for(j=1,i, t=u[i]*u[j]; if(t>lim,break); listput(v,t))); Set(v) \\ Charles R Greathouse IV, May 13 2015

A072544 Numbers whose smallest decimal digit is also the initial digit.

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, 121, 122, 123, 124

Offset: 1

Reinhard Zumkeller, Aug 04 2002

A054054(a(n)) = A000030(a(n));

the sequence differs from A009994, A032857 and A032898: a(65)=121 is not in A009994, a(58)=113 is not in A032857 and a(56)=111 is not in A032898.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A072543, A009994.

a072544 n = a072544_list !! (n-1)
a072544_list = [x | x <- [0..], a054054 x == a000030 x]
-- Reinhard Zumkeller, Apr 25 2012

sddiQ[n_]:=Module[{idn=IntegerDigits[n]},First[idn]==Min[idn]]; Select[ Range[ 0,130],sddiQ] (* Harvey P. Dale, Oct 30 2011 *)

A085932 Numbers k such that (digits of k sorted in ascending order) + (digital sum of k) is a palindrome.

Original entry on oeis.org

1, 2, 3, 4, 10, 20, 30, 40, 100, 124, 129, 142, 148, 167, 176, 184, 192, 200, 214, 219, 224, 229, 241, 242, 248, 267, 276, 284, 291, 292, 300, 348, 367, 376, 384, 400, 412, 418, 421, 422, 428, 438, 448, 467, 476, 481, 482, 483, 484, 567, 576, 617, 627, 637

Offset: 1

Jason Earls and Amarnath Murthy, Jul 14 2003

Essentially all terms can be generated by going over A009994. By permuting digits and including any number of 0's in any term that is in A009994 any term in this sequence can be found. For example, from 124 we find that 412, 1402, 200004001 are terms. - David A. Corneth, Apr 20 2024

			142 is a term because the digits of 142 in ascending order are 124, the digital sum of 124 is 7, and 124 + 7 = 131, a palindrome.

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A009994, A085933, A085934, A085935.

dspQ[n_]:=Module[{sidn=Sort[IntegerDigits[n]],pidn},pidn= IntegerDigits[ FromDigits[ sidn]+ Total[ sidn]]; pidn==Reverse[pidn]]; Select[Range[ 700], dspQ] (* Harvey P. Dale, Jul 19 2011 *)

A277061 Numbers with multiplicative digital root > 0.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 26, 27, 28, 29, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 46, 47, 48, 49, 51, 53, 57, 61, 62, 63, 64, 66, 67, 68, 71, 72, 73, 74, 75, 76, 77, 79, 81, 82, 83, 84, 86, 88, 89, 91, 92, 93, 94, 97, 98, 99, 111, 112, 113, 114, 115

Offset: 1

J. Lowell, Sep 26 2016

Question: when will numbers not in this sequence outnumber numbers in this sequence? Up to n = 1249, there are 524 terms, so 525 terms not in this sequence. Up to n = 1522, there are n/2 terms. No n > 1522 has that property. Up to 10^10, only about 1.46% of numbers are a term.

To find how many terms there are up to 10^n, see if A009994(i) is for 2 <= i <= binomial(n + 9, 9). If it is then that adds A047726(A009994(i)) to the total (we don't have to worry about digits 0 in A009994(i) as there aren't any for the specified i). One may put further constraints on i. For example, A009994(i) can't contain an even digit and a 5 in the same number. - David A. Corneth, Sep 27 2016

			25 is not in this sequence because 2*5 = 10 and 1*0 = 0.

Index entries for 10-automatic sequences.

Cf. A009994, A047726.
Cf. A031347, A034048 (complement).
Cf. A028843 (a subsequence).
Union of A002275, A034049, A034050, A034051, A034052, A034053, A034054, A034055, A034056 (numbers having multiplicative digital roots 1-9).
Cf. A052382 (a supersequence).

Select[Range@ 112, FixedPoint[Times @@ IntegerDigits@ # &, #] > 0 &] (* Michael De Vlieger, Sep 26 2016 *)

is(n) = n=digits(n); while(#n>1,n=digits(prod(i=1,#n,n[i]))); #n>0 \\ David A. Corneth, Sep 27 2016

More terms from Michael De Vlieger, Sep 26 2016

A355062 Perfect powers whose digits are in nondecreasing order.

Original entry on oeis.org

4, 8, 9, 16, 25, 27, 36, 49, 125, 128, 144, 169, 225, 256, 289, 1156, 1225, 1369, 1444, 4489, 6889, 11236, 11449, 13456, 13689, 27889, 33489, 111556, 112225, 113569, 134689, 146689, 344569, 444889, 2666689, 2778889, 11115556, 11122225, 11135569, 11336689

Offset: 1

Jon E. Schoenfield, Jun 16 2022

I.e., numbers of the form b^k with b > 1 and k > 1 in whose base-10 expansion no digit is less than the previous digit.

Includes infinite subsequences such as {16, 1156, 111556, 11115556, ...} and {25, 1225, 112225, 11122225, ...}, so the sequence is infinite.

Michael S. Branicky, Table of n, a(n) for n = 1..261

Cf. A001597, A009994, A355060.

perfectPowerQ[n_] := GCD @@ FactorInteger[n][[All, 2]] > 1; (* A001597 *) Select[Range[10^6], perfectPowerQ[#] && Min[Differences[IntegerDigits[#]]]>-1 &] (* Stefano Spezia, Jul 01 2025 *)

isok(m) = if (ispower(m), my(d=digits(m)); (d == vecsort(d))); \\ Michel Marcus, Jun 18 2022

from sympy import perfect_power as pp
from itertools import count, islice, combinations_with_replacement as mc
def agen():
    for d in count(1):
        ni = (int("".join(m)) for m in mc("123456789", d))
        yield from filter(pp, ni)
print(list(islice(agen(), 40))) # Michael S. Branicky, Jun 16 2022

A355222 The k-th leftmost digit of a(n) is the greatest of the k leftmost digits of n.

Original entry on oeis.org

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

Offset: 0

Rémy Sigrist, Jun 24 2022

Leading zeros are ignored.

			For n = 1402: max({1}) = 1, max({1, 4}) = 4, max({1, 4, 0}) = 4, max({1, 4, 0, 2}) = 4, so a(1402) = 1444.

Rémy Sigrist, Table of n, a(n) for n = 0..10000

See A355221, A355223 and A355224 for similar sequences.

Cf. A003817 (binary analog), A009994 (fixed points).

a(n, base=10) = { my (d=digits(n, base), m=-oo); for (k=1, #d, d[k]=m=max(m, d[k])); fromdigits(d, base) }

def a(n):
    s, m = str(n), "0"
    return int("".join((m:=max(m, s[k])) for k in range(len(s))))
print([a(n) for n in range(68)]) # Michael S. Branicky, Jun 24 2022

from itertools import accumulate
def A355222(n): return int(''.join(accumulate(str(n),func=max))) # Chai Wah Wu, Jun 25 2022

a(n) >= n with equality iff n belongs to A009994.

a(a(n)) = a(n).

A355223 The k-th rightmost digit of a(n) is the least of the k rightmost digits of n.

Original entry on oeis.org

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

Offset: 0

Rémy Sigrist, Jun 24 2022

Leading zeros are ignored.

			For n = 1402:
- min({1, 4, 0, 2}) = 0,
- min({4, 0, 2}) = 0,
- min({0, 2}) = 0,
- min({2}) = 2,
- so a(1402) = 2.

Rémy Sigrist, Table of n, a(n) for n = 0..10000

See A355221, A355222 and A355224 for similar sequences.

Cf. A008592, A009994 (fixed points), A135481 (binary analog).

a(n, base=10) = { my (d=digits(n, base), m=oo); forstep (k=#d, 1, -1, d[k]=m=min(m, d[k])); fromdigits(d, base) }

def a(n):
    s, m = str(n), "9"
    return int("".join((m:=min(m, s[-1-k])) for k in range(len(s)))[::-1])
print([a(n) for n in range(69)]) # Michael S. Branicky, Jun 24 2022

from itertools import accumulate
def A355223(n): return int(''.join(accumulate(str(n)[::-1],func=min))[::-1]) # Chai Wah Wu, Jun 25 2022

a(n) <= n with equality iff n belongs to A009994.
a(a(n)) = a(n).
a(n) = 0 iff n is a multiple of 10.

cp's OEIS Frontend

A329299 Numbers whose digits are in nondecreasing order in bases 9 and 10.

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A028820 Squares with digits in nondecreasing order.

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A193581 Sort-and-subtract: a(n) = n - A004185(n).

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Haskell

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A254143 Products of any two not necessarily distinct terms of A237424.

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Haskell

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A072544 Numbers whose smallest decimal digit is also the initial digit.

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Haskell

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A085932 Numbers k such that (digits of k sorted in ascending order) + (digital sum of k) is a palindrome.

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Mathematica

A277061 Numbers with multiplicative digital root > 0.

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Mathematica

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A355062 Perfect powers whose digits are in nondecreasing order.

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