A383062 -id:A383062 - cp's OEIS frontend

A382465 Positive integers such that every even digit except the first is immediately preceded by a smaller digit.

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 23, 24, 25, 26, 27, 28, 29, 31, 33, 34, 35, 36, 37, 38, 39, 41, 43, 45, 46, 47, 48, 49, 51, 53, 55, 56, 57, 58, 59, 61, 63, 65, 67, 68, 69, 71, 73, 75, 77, 78, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99

Offset: 1

Paolo Xausa, Mar 28 2025

Conjecture: these are the terms of A382462, sorted.

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. A382462, A382464 (complement).

Cf. A377912, A382624, A382938, A383062, A383246, A383248, A383250, A383501.

A382465Q[k_] := FreeQ[Partition[IntegerDigits[k], 2, 1], {i_, j_?EvenQ} /; i >= j];
Select[Range[100], A382465Q]

def ok(n):
    s = str(n)
    return n and all(d not in "02468" or s[i-1] 0)
print([k for k in range(100) if ok(k)]) # Michael S. Branicky, Apr 30 2025

A383061 Positive integers that contain an odd digit d immediately preceded by a digit >= d.

Original entry on oeis.org

11, 21, 31, 33, 41, 43, 51, 53, 55, 61, 63, 65, 71, 73, 75, 77, 81, 83, 85, 87, 91, 93, 95, 97, 99, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 121, 131, 133, 141, 143, 151, 153, 155, 161, 163, 165, 171, 173, 175, 177, 181, 183, 185, 187, 191, 193, 195, 197, 199

Offset: 1

Paolo Xausa, Apr 18 2025

Conjecture: these are the numbers missing from A383059.

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. A383059, A383062 (complement).

Cf. A347298, A382464, A382623, A382937, A383245, A383247, A383249, A383500.

A383061Q[k_] := MemberQ[Partition[IntegerDigits[k], 2, 1], {i_, j_?OddQ} /; i >= j];
Select[Range[200], A383061Q]

def ok(n):
    s = str(n)
    return any(s[i] <= s[i-1] for i in range(1, len(s)) if s[i] in "13579")
print([k for k in range(200) if ok(k)]) # Michael S. Branicky, Apr 19 2025

A382938 Nonnegative integers such that every odd digit except the leftmost is immediately preceded by a larger digit.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 16, 18, 20, 21, 22, 24, 26, 28, 30, 31, 32, 34, 36, 38, 40, 41, 42, 43, 44, 46, 48, 50, 51, 52, 53, 54, 56, 58, 60, 61, 62, 63, 64, 65, 66, 68, 70, 71, 72, 73, 74, 75, 76, 78, 80, 81, 82, 83, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95

Offset: 1

Paolo Xausa, Apr 14 2025

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. A382937 (complement).

Cf. A377912, A382465, A382624, A383062, A383246, A383248, A383250, A383501.

A382938Q[k_] := FreeQ[Partition[IntegerDigits[k], 2, 1], {i_, j_?OddQ} /; i <= j];
Select[Range[0, 100], A382938Q]

def ok(n):
    s = str(n)
    return all(s[i+1] < s[i] for i in range(len(s)-1) if s[i+1] in "13579")
print([k for k in range(96) if ok(k)]) # Michael S. Branicky, Apr 14 2025

A383246 Positive integers without the digit 0 such that every even digit except the rightmost is immediately followed by a smaller digit.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 71, 72, 73, 74, 75, 76, 77, 78, 79, 81, 82, 83, 84, 85, 86, 87, 91, 92, 93, 94, 95, 96, 97, 98, 99

Offset: 1

Paolo Xausa, Apr 20 2025

Conjecture: these are the terms of A342043, sorted.

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. A342043, A383245 (complement).

Cf. A377912, A382465, A382624, A382938, A383062, A383248, A383250, A383501.

A383246Q[k_] := FreeQ[#, 0] && FreeQ[Partition[#, 2, 1], {i_?EvenQ, j_} /; j >= i] &  [IntegerDigits[k]];
Select[Range[200], A383246Q]

def ok(n):
    s = str(n)
    return "0" not in s and all(d not in "02468" or s[i]Michael S. Branicky, Apr 28 2025

A383248 Nonnegative integers without the digit 9 such that every odd digit except the rightmost is immediately followed by a larger digit.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 23, 24, 25, 26, 27, 28, 34, 35, 36, 37, 38, 40, 41, 42, 43, 44, 45, 46, 47, 48, 56, 57, 58, 60, 61, 62, 63, 64, 65, 66, 67, 68, 78, 80, 81, 82, 83, 84, 85, 86, 87, 88, 120, 121, 122, 123, 124, 125, 126, 127, 128

Offset: 1

Paolo Xausa, Apr 25 2025

Conjecture: these are the terms of A342044, sorted.

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. A342044, A383247 (complement).

Cf. A377912, A382465, A382624, A382938, A383062, A383246, A383250.

A383248Q[k_] := FreeQ[#, 9] && FreeQ[Partition[#, 2, 1], {i_?OddQ, j_} /; j <= i] & [IntegerDigits[k]];
Select[Range[0, 200], A383248Q]

def ok(n):
    s = str(n)
    return "9" not in s and all(d not in "13579" or s[i]>d for i, d in enumerate(s, 1) if i < len(s))
print([k for k in range(129) if ok(k)]) # Michael S. Branicky, Apr 28 2025

A383250 Nonnegative integers not ending with the digit 1 and such that every odd digit except the rightmost is immediately followed by a smaller digit.

Original entry on oeis.org

0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 20, 22, 23, 24, 25, 26, 27, 28, 29, 30, 32, 40, 42, 43, 44, 45, 46, 47, 48, 49, 50, 52, 53, 54, 60, 62, 63, 64, 65, 66, 67, 68, 69, 70, 72, 73, 74, 75, 76, 80, 82, 83, 84, 85, 86, 87, 88, 89, 90, 92, 93, 94, 95, 96, 97, 98, 100, 102, 103

Offset: 1

Paolo Xausa, Apr 26 2025

Conjecture: these are the terms of A342045, sorted.
From Quinn Savitt, Apr 29 2025: (Start)
Theorem: These are the terms of A342045, sorted.
The proof formally defines an extendibility condition: a finite set of selected numbers remains extendible if, for every finite subset, there exists a new number satisfying the "odd digit implies next smaller digit" rule and not ending in 1.
Using induction and monotonicity of extendibility, it follows that every number satisfying these conditions eventually appears in the greedy construction of A342045. This implies that the sequence is a sorted version of A342045. (End)

Paolo Xausa, Table of n, a(n) for n = 1..10000
Quinn Savitt, Formal proof that A383250 = sorted(A342045)

Cf. A342045, A383249 (complement).

Cf. A377912, A382465, A382624, A383062, A382938, A383246, A383248.

A383250Q[k_] := Last[#] != 1 && FreeQ[Partition[#, 2, 1], {i_?OddQ, j_} /; j >= i] & [IntegerDigits[k]];
Select[Range[0, 200], A383250Q]

def ok(n):
    if n%10 == 1: return False
    s = str(n)
    return all(d in "02468" or s[i]Michael S. Branicky, Apr 28 2025

A383059 Lexicographically earliest sequence of distinct nonnegative integers such that if a digit d in the digit stream (ignoring commas) is odd, the previous digit is < d.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 20, 10, 12, 22, 23, 24, 25, 26, 27, 28, 29, 40, 13, 42, 30, 14, 44, 45, 46, 47, 48, 49, 60, 15, 62, 32, 34, 50, 16, 64, 52, 35, 66, 67, 68, 69, 80, 17, 82, 36, 70, 18, 84, 54, 56, 72, 37, 86, 74, 57, 88, 89, 200, 19, 201, 38, 90, 39, 202, 58

Offset: 1

Paolo Xausa, Apr 18 2025

Could be summarized as "odd digit, previous smaller". A variant of A342042.

Paolo Xausa, Table of n, a(n) for n = 1..10000
Paolo Xausa, Color scatterplot of the first 100000 terms

Similar sequences: A342042, A342043, A342044, A342045, A382462, A382621, A382935.

Cf. A383060, A383061, A383062, A383063.

A383059list[nmax_] := Module[{a, s, invQ, fu = 1},
  invQ[k_] := invQ[k] = (If[#, s[k] = #]; #) & [MemberQ[Partition[IntegerDigits[k], 2, 1], {i_, j_?OddQ} /; i >= j]];
  s[_] := False; s[0] = True;
  NestList[(a = fu; While[s[a] || invQ[a] || invQ[# + First[IntegerDigits[a]]], a++] & [Mod[#, 10]*10]; While[s[fu], fu++]; s[a] = True; a) &, 0, nmax - 1]];
A383059list[100]

from itertools import count, islice
def cond(s):
    return all(s[i] > s[i-1] for i in range(1, len(s)) if s[i] in "13579")
def agen(): # generator of terms
    an, seen, s, m = 0, {0}, "0", 1
    while True:
        yield an
        an = next(k for k in count(m) if k not in seen and cond(s[-1]+str(k)))
        seen.add(an); s += str(an)
        while m in seen or not cond(str(m)): m += 1
print(list(islice(agen(), 70))) # Michael S. Branicky, Apr 19 2025

A383501 Nonnegative integers without the digit 9 such that every odd digit except the leftmost is immediately preceded by a larger digit.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 16, 18, 20, 21, 22, 24, 26, 28, 30, 31, 32, 34, 36, 38, 40, 41, 42, 43, 44, 46, 48, 50, 51, 52, 53, 54, 56, 58, 60, 61, 62, 63, 64, 65, 66, 68, 70, 71, 72, 73, 74, 75, 76, 78, 80, 81, 82, 83, 84, 85, 86, 87, 88, 100, 102, 104, 106, 108

Offset: 1

Paolo Xausa, Apr 29 2025

Conjecture: these are the terms of A382935, sorted.

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. A382935, A383500 (complement).

Cf. A377912, A382465, A382624, A382938, A383062, A383246, A383248, A383250.

A383501Q[k_] := FreeQ[#, 9] && FreeQ[Partition[#, 2, 1], {i_, j_?OddQ} /; i <= j] & [IntegerDigits[k]];
Select[Range[0, 200], A383501Q]

def ok(n):
    s = str(n)
    return "9" not in s and all(d not in "13579" or s[i-1]>d for i, d in enumerate(s) if i > 0)
print([k for k in range(134) if ok(k)]) # Michael S. Branicky, Apr 29 2025

cp's OEIS Frontend

A382465 Positive integers such that every even digit except the first is immediately preceded by a smaller digit.

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A383061 Positive integers that contain an odd digit d immediately preceded by a digit >= d.

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A382938 Nonnegative integers such that every odd digit except the leftmost is immediately preceded by a larger digit.

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A383246 Positive integers without the digit 0 such that every even digit except the rightmost is immediately followed by a smaller digit.

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A383248 Nonnegative integers without the digit 9 such that every odd digit except the rightmost is immediately followed by a larger digit.

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A383250 Nonnegative integers not ending with the digit 1 and such that every odd digit except the rightmost is immediately followed by a smaller digit.

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A383059 Lexicographically earliest sequence of distinct nonnegative integers such that if a digit d in the digit stream (ignoring commas) is odd, the previous digit is < d.

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A383501 Nonnegative integers without the digit 9 such that every odd digit except the leftmost is immediately preceded by a larger digit.

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Keywords

Comments

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Programs

Mathematica

Python