A382462 -id:A382462 - cp's OEIS frontend

A382463 First differences of A382462.

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 12, -10, 2, 10, 1, 1, 1, 1, 1, 1, 12, -27, 29, -12, -16, 30, 1, 1, 1, 1, 12, -45, 47, -30, 2, 16, -34, 48, -12, -17, 31, 1, 1, 12, -63, 65, -46, 34, -52, 66, -30, 2, 16, -35, 49, -12, -17, 31, 12, -62, 63, -43, 44

Offset: 1

Paolo Xausa, Mar 28 2025

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

Cf. A382462.

(* A382462list is defined at A382462 *)
Differences[A382462list[100]]

A382466 Split A382462 into runs of increasing elements. a(n) is the length of the n-th run.

Original entry on oeis.org

19, 10, 2, 1, 7, 2, 3, 2, 1, 5, 2, 2, 2, 3, 2, 1, 3, 2, 2, 61, 11, 9, 8, 8, 7, 7, 6, 6, 3, 2, 4, 8, 2, 8, 2, 7, 2, 6, 2, 6, 2, 1, 8, 2, 5, 2, 2, 2, 1, 2, 1, 6, 2, 2, 1, 1, 2, 8, 6, 2, 6, 3, 2, 1, 5, 2, 2, 1, 4, 2, 1, 2, 6, 6, 3, 2, 2, 2, 2, 1, 2, 1, 2, 1, 1, 5, 5, 4, 2, 2, 4

Offset: 1

Paolo Xausa, Mar 28 2025

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

Cf. A382462.

(* A382462list is defined at A382462 *)
Map[Length, Most[Split[A382462list[500], Less]]]

A382464 Positive integers that contain an even digit d immediately preceded by a digit >= d.

Original entry on oeis.org

10, 20, 22, 30, 32, 40, 42, 44, 50, 52, 54, 60, 62, 64, 66, 70, 72, 74, 76, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 120, 122, 130, 132, 140, 142, 144, 150, 152, 154, 160, 162, 164, 166, 170, 172, 174, 176, 180

Offset: 1

Paolo Xausa, Mar 28 2025

Conjecture: these are the numbers missing from A382462.

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

Cf. A382462, A382465 (complement).

Cf. A347298, A382623, A382937, A383061, A383245, A383247, A383249, A383500.

A382464Q[k_] := MemberQ[Partition[IntegerDigits[k], 2, 1], {i_, j_?EvenQ} /; i >= j];
Select[Range[200], A382464Q]

def ok(n):
    s = str(n)
    return any(d in "02468" and s[i-1]>=d for i, d in enumerate(s) if i > 0)
print([k for k in range(181) if ok(k)]) # Michael S. Branicky, Apr 30 2025

A382465 Positive integers such that every even digit except the first is immediately preceded 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, 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

A382621 Lexicographically earliest sequence of distinct positive integers such that if a digit d in the digit stream (ignoring commas) is even, the previous digit is > d.

Original entry on oeis.org

1, 3, 2, 5, 4, 7, 6, 9, 8, 10, 11, 13, 15, 17, 19, 20, 30, 31, 32, 33, 21, 35, 23, 25, 27, 29, 37, 39, 40, 50, 51, 52, 53, 54, 55, 41, 57, 42, 59, 43, 70, 71, 72, 73, 74, 75, 45, 47, 49, 60, 76, 77, 61, 79, 62, 90, 91, 92, 93, 94, 95, 96, 97, 63, 98, 64, 99, 65, 101, 103

Offset: 1

Paolo Xausa, Apr 01 2025

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

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

Cf. A342042, A382462, A382622, A382623, A382624, A382625.

A382621list[nmax_] := Module[{a, s, invQ, fu = 2},
  invQ[k_] := invQ[k] = (If[#, s[k] = #]; #) & [MemberQ[Partition[IntegerDigits[k], 2, 1], {i_, j_?EvenQ} /; i <= j]];
  s[_] := False; s[1] = 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) &, 1, nmax - 1]];
A382621list[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 "02468")
def agen(): # generator of terms
    an, seen, s, m = 1, {1}, "1", 2
    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 03 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

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

cp's OEIS Frontend

A382463 First differences of A382462.

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A382466 Split A382462 into runs of increasing elements. a(n) is the length of the n-th run.

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A382464 Positive integers that contain an even digit d immediately preceded by a digit >= d.

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

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A382621 Lexicographically earliest sequence of distinct positive integers such that if a digit d in the digit stream (ignoring commas) is even, the previous digit is > d.

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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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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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Mathematica

Python