A383501 -id:A383501 - 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

A383062 Nonnegative integers such that every odd digit except the leftmost is immediately preceded by a smaller digit.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27, 28, 29, 30, 32, 34, 35, 36, 37, 38, 39, 40, 42, 44, 45, 46, 47, 48, 49, 50, 52, 54, 56, 57, 58, 59, 60, 62, 64, 66, 67, 68, 69, 70, 72, 74, 76, 78, 79, 80, 82, 84, 86, 88, 89, 90

Offset: 1

Paolo Xausa, Apr 18 2025

Conjecture: these are the terms of A383059, sorted.

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

Cf. A383059, A383061 (complement).

Cf. A377912, A382465, A382624, A382938, A383246, A383248, A383250, A383501.

A383062Q[k_] := FreeQ[Partition[IntegerDigits[k], 2, 1], {i_, j_?OddQ} /; i >= j];
Select[Range[0, 100], A383062Q]

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(91) 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

A382935 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, 2, 1, 4, 3, 6, 5, 8, 7, 10, 20, 21, 22, 12, 14, 16, 18, 24, 26, 28, 30, 40, 41, 42, 43, 44, 31, 46, 32, 48, 34, 36, 38, 50, 60, 61, 62, 63, 64, 65, 66, 51, 68, 52, 80, 81, 82, 83, 84, 85, 86, 53, 87, 54, 88, 56, 58, 70, 200, 202, 100, 204, 102, 104, 106, 108, 71, 206, 120, 208

Offset: 1

Paolo Xausa, Apr 14 2025

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

No term contains the digit 9.

Paolo Xausa, Table of n, a(n) for n = 1..10000
Paolo Xausa, Color scatterplot of the first 50000 terms

Similar sequences: A342042, A342043, A342044, A342045, A382462, A382621, A383059.

Cf. A382936, A382939, A383500, A383501.

A382935list[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++] & [Max[Mod[#, 10], 1]*10]; While[s[fu], fu++]; s[a] = True; a) &, 0, nmax - 1]];
A382935list[100]

from itertools import count, islice
def cond(s):
    return all(s[i+1] < s[i] for i in range(len(s)-1) if s[i+1] 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 14 2025

A383500 Positive integers that contain the digit 9, or an odd digit d immediately preceded by a digit <= d.

Original entry on oeis.org

9, 11, 13, 15, 17, 19, 23, 25, 27, 29, 33, 35, 37, 39, 45, 47, 49, 55, 57, 59, 67, 69, 77, 79, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 101, 103, 105, 107, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 123, 125, 127, 129, 130, 131, 132, 133, 134, 135, 136, 137

Offset: 1

Paolo Xausa, Apr 29 2025

Conjecture: these are the numbers missing from A382935.

Theorem: This sequence is the complement of A382935. - Quinn Savitt, May 08 2025

Paolo Xausa, Table of n, a(n) for n = 1..10000
Quinn Savitt, Proof that A383500 is the Complement of A382935

Cf. A382935, A383501 (complement).

Cf. A347298, A382464, A382623, A382937, A383061, A383245, A383247, A383249.

A383500Q[k_] := MemberQ[#, 9] || MemberQ[Partition[#, 2, 1], {i_, j_?OddQ} /; i <= j] & [IntegerDigits[k]];
Select[Range[200], A383500Q]

def ok(n):
    s = str(n)
    return "9" in s or any(d in "13579" and 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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A383062 Nonnegative integers such that every odd digit except the leftmost is immediately preceded by a smaller digit.

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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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A382935 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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A383500 Positive integers that contain the digit 9, or an odd digit d immediately preceded by a digit <= d.

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Python