A009993 -id:A009993 - cp's OEIS frontend

A131058 List of numbers n whose reverse, R(n), is not less than n.

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, 101, 102, 103, 104, 105, 106, 107, 108, 109, 111, 112, 113, 114, 115

Offset: 1

Zak Seidov, Sep 24 2007

First terms not in A009993 are 11, 22, 33, etc.

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

Cf. A002113, A009993.

Select[Range[0, 200], FromDigits[Reverse[IntegerDigits[ # ]]]>=#&]
Select[Range[0,200],IntegerReverse[#]>=#&]

is(n) = my(d=digits(n));sum(i=1,#d,10^(#d-i)*d[i]) <= sum(i=1,#d,10^(i-1)*d[i]) \\ David A. Corneth, Apr 25 2016

isok(n) = subst(Polrev(digits(n)), x, 10) >= n; \\ Michel Marcus, Apr 25 2016

A135355 Semiprimes with digits in ascending order.

Original entry on oeis.org

4, 6, 9, 14, 15, 25, 26, 34, 35, 38, 39, 46, 49, 57, 58, 69, 123, 129, 134, 145, 146, 158, 159, 169, 178, 235, 237, 247, 249, 259, 267, 278, 289, 346, 358, 458, 469, 478, 489, 579, 589, 679, 689, 789, 1234, 1238, 1247, 1257, 1267, 1345, 1346, 1347, 1349, 1357

Offset: 1

Jonathan Vos Post, Dec 08 2007

This is to A028864 as A001358 is to A000040. Semiprime metadromes in base 10. Contains 175 terms, the last being a(175)=12345789 = 3 * 4115263.

Snapshots: a(100)=13579, a(150)=135689, a(160)=245678, a(163)=356789, a(169)=1346789, a(173)=2356789. - R. J. Mathar, Dec 11 2007

R. J. Mathar, Table of n, a(n) for n = 1..175

Cf. A001358, A028864.

isA001358 := proc(n) if numtheory[bigomega](n) = 2 then true; else false ; fi ; end: isA009993 := proc(n) local dgs,i ; dgs := convert(n,base,10) ; if nops(dgs) = 1 then RETURN(true) ; fi ; for i from 1 to nops(dgs)-1 do if op(i,dgs) <= op(i+1,dgs) then RETURN(false) ; fi ; od: RETURN(true) ; end: isA135355 := proc(n) isA001358(n) and isA009993(n) ; end: for n from 4 to 1400 do if isA135355(n) then printf("%d, ",n) ; fi ; od: # R. J. Mathar, Dec 11 2007
# second Maple program:
b:= proc(n) n, seq(b(10*n+j), j=irem(n, 10)+1..9) end:
select(numtheory[bigomega]=2, {seq(b(n), n=1..9)})[];  # Alois P. Heinz, Apr 13 2025

Select[Range[1357],PrimeOmega[#]==2&&AllTrue[Differences[IntegerDigits[#]],Positive]&] (* James C. McMahon, Apr 13 2025 *)

Equals A001358 INTERSECT A009993. - R. J. Mathar, Dec 11 2007

Corrected by R. J. Mathar, Dec 11 2007

A211771 Nonprime numbers with distinct digits in ascending order.

Original entry on oeis.org

1, 4, 6, 8, 9, 12, 14, 15, 16, 18, 24, 25, 26, 27, 28, 34, 35, 36, 38, 39, 45, 46, 48, 49, 56, 57, 58, 68, 69, 78, 123, 124, 125, 126, 128, 129, 134, 135, 136, 138, 145, 146, 147, 148, 156, 158, 159, 168, 169, 178, 189, 234, 235, 236, 237, 238, 245, 246, 247

Offset: 1

Jaroslav Krizek, May 07 2012

Sequence is finite with 411 terms, last term is a(411) = 123456789.

Complement of A052015 with respect to A009993. Supersequence of A211772.

Jaroslav Krizek, Table of n, a(n) for n = 1..411 (complete list)

Cf. A052015 (primes with distinct digits in ascending order), A009993 (numbers with distinct digits in ascending order), A211772 (nonprime numbers all of whose divisors are numbers whose decimal digits are in ascending order).

nnddQ[n_]:=!PrimeQ[n]&&Max[DigitCount[n]]<2&&Min[Differences[ IntegerDigits[ n]]]>0; Select[Range[300],nnddQ] (* Harvey P. Dale, Sep 27 2020 *)

A178788(a(n)) = 1.

A377948 Numbers that have at least 1 repeated decimal digit and whose decimal digits are nondecreasing as place value decreases.

Original entry on oeis.org

11, 22, 33, 44, 55, 66, 77, 88, 99, 111, 112, 113, 114, 115, 116, 117, 118, 119, 122, 133, 144, 155, 166, 177, 188, 199, 222, 223, 224, 225, 226, 227, 228, 229, 233, 244, 255, 266, 277, 288, 299, 333, 334, 335, 336, 337, 338, 339, 344, 355, 366, 377, 388, 399

Offset: 1

Michael De Vlieger, Nov 14 2024

Intersection of A009994 and A109303.

Does not intersect either A009993 or A009995.

Michael De Vlieger, Table of n, a(n) for n = 1..10938 (includes all numbers up to 7 digits long)

Cf. A009993, A009994, A009995, A109303, A178788.

Select[Range[10^6], And[CountDistinct[#] != Length[#], AllTrue[Differences[#], # >= 0 &]] &[IntegerDigits[#]] &]
(* More efficient program: *)
b = 10; mm = b - 1; nn = 14;
s = Table[Map[Position[#, 1][[All, 1]] &,
  Permutations@ Join[ConstantArray[1, r], ConstantArray[0, mm - r] ] ],
    {r, Min[mm, nn]}];
Union@ Flatten@ Table[
  w = Apply[Join, Permutations /@ IntegerPartitions[n, Min[mm, n - 1] ] ];
  Reap[Do[
    Sow[Table[FromDigits[Flatten@
      MapIndexed[ConstantArray[m[[First[#2] ]], #1] &,
      w[[i]]], b], {m, s[[Length[w[[i]]] ]]} ] ],
    {i, Length[w]}] ][[-1, 1]], {n, 2, nn}]

A178788(a(n)) = 0.

A118629 Number of n-digit integers with decimal digits in increasing order.

Original entry on oeis.org

10, 36, 84, 126, 126, 84, 36, 9, 1

Offset: 1

Zak Seidov, May 09 2006

There are precisely 512 such numbers, see A009993.

			a(2)=36 because there are 36 two-digit integers whose decimal digits are in increasing order (see A009993): 12, 13, 14, 15, 16, 17, 18, 19, 23, 24, 25, 26, 27, 28, 29, 34, 35, 36, 37, 38, 39, 45, 46, 47, 48, 49, 56, 57, 58, 59, 67, 68, 69, 78, 79, 89

Cf. A009993.

a(1)=1, at n>1: a(n)=binomial(10, n)(1 - n/10)

Edited by Charles R Greathouse IV, Aug 05 2010

A119260 Numbers with even decimal digits in increasing order.

Original entry on oeis.org

0, 2, 4, 6, 8, 24, 26, 28, 46, 48, 68, 246, 248, 268, 468, 2468

Offset: 1

Zak Seidov, May 11 2006

This is the complete list of all 16 such numbers. Cf. A119261 Even decimal digits in decreasing order, A119253 Odd digits in increasing order, A119252 Odd digits in decreasing order, A009993 Digits in increasing order, A009995 Digits in decreasing order.

Cf. A009993, A009995, A119252, A119253, A119261.

Flatten@Table[FromDigits/@Subsets[Range[2,8,2],{n}],{n,0,5}]

A119261 Numbers with even decimal digits in decreasing order.

Original entry on oeis.org

0, 2, 4, 6, 8, 20, 40, 42, 60, 62, 64, 80, 82, 84, 86, 420, 620, 640, 642, 820, 840, 842, 860, 862, 864, 6420, 8420, 8620, 8640, 8642, 86420

Offset: 1

Zak Seidov, May 11 2006

This is the complete list of all 31 such numbers. Cf. A119260 Even decimal digits in increasing order, A119253 Odd digits in increasing order, A119252 Odd digits in decreasing order, A009993 Digits in increasing order, A009995 Digits in decreasing order.

Cf. A009993, A009995, A119252, A119253, A119260.

Sort@Flatten@Table[FromDigits/@Subsets[Range[8,0,-2],{n}],{n,5}]

A295638 Take the sequence of nonnegative integers whose decimal digits are not in strictly increasing order. Partition the sequence into subsequences whose elements are consecutive integers. Then a(n) is the number of elements in the n-th partition.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 33, 4, 5, 6, 7, 8, 9, 44, 5, 6, 7, 8, 9, 55, 6, 7, 8, 9, 66, 7, 8, 9, 77, 8, 9, 88, 9, 99, 444, 5, 6, 7, 8, 9, 55, 6, 7, 8, 9, 66, 7, 8, 9, 77, 8, 9, 88, 9, 99, 555, 6, 7, 8, 9, 66, 7, 8, 9, 77, 8, 9, 88, 9, 99, 666, 7, 8

Offset: 1

Gunnar Lee Johnson, Nov 24 2017

Only defined as an integer for a(1) through a(255), as a(256) references the infinite partition (123456790, 123456791, ..., 999999999, 1000000000, 1000000001, ...). No integer greater than 123456789 has a strictly increasing sequence of digits (itself being the only case for 9 digits, and by the pigeonhole principle, a >9-digit number must have a digit repeated and is thus not strictly increasing).

			For a(1)=2 through a(8)=9, these correspond to the consecutive subsequences (10, 11), (20, 21, 22), ..., (80, 81, 82, ..., 88). The jumps at e.g. a(9)=33 or a(37)=444 correspond to (90, 91, ..., 122) and (790, 791, ..., 1233), where 89 and 123, and 789 and 1234, are the values partitioning the subsequences.

Gunnar Lee Johnson, Table of n, a(n) for n = 1..255

The nonnegative integers minus A009993 is the sequence that is partitioned.

is(n) = my(d=digits(n)); d != vecsort(d,,8);
lista(nn) = {my(w = select(n->is(n), vector(nn, k, k))); my(dw = vector(#w-1, k, w[k+1] - w[k])); my(k = 1); for (n=1, #dw, if (dw[n] == 1, k++, print1(k, ", "); k = 1););} \\ Michel Marcus, Jan 08 2018

def a(n):
    (x,i,count,switch) = (0,0,1,True)
    while True:
        if switch == (list(sorted(set(str(i)))) == list(str(i))):
            count += 1
        else:
            if not switch: x += 1
            if x == n: return count
            (count, switch) = (1, not switch)
        i += 1

cp's OEIS Frontend

A131058 List of numbers n whose reverse, R(n), is not less than n.

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A135355 Semiprimes with digits in ascending order.

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A211771 Nonprime numbers with distinct digits in ascending order.

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A377948 Numbers that have at least 1 repeated decimal digit and whose decimal digits are nondecreasing as place value decreases.

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A118629 Number of n-digit integers with decimal digits in increasing order.

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A119260 Numbers with even decimal digits in increasing order.

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A119261 Numbers with even decimal digits in decreasing order.

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A295638 Take the sequence of nonnegative integers whose decimal digits are not in strictly increasing order. Partition the sequence into subsequences whose elements are consecutive integers. Then a(n) is the number of elements in the n-th partition.

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