A382621 -id:A382621 - cp's OEIS frontend

A382622 First differences of A382621.

2, -1, 3, -1, 3, -1, 3, -1, 2, 1, 2, 2, 2, 2, 1, 10, 1, 1, 1, -12, 14, -12, 2, 2, 2, 8, 2, 1, 10, 1, 1, 1, 1, 1, -14, 16, -15, 17, -16, 27, 1, 1, 1, 1, 1, -30, 2, 2, 11, 16, 1, -16, 18, -17, 28, 1, 1, 1, 1, 1, 1, 1, -34, 35, -34, 35, -34, 36, 2, 2, 2, -40, 2, 11, 29

Offset: 1

Paolo Xausa, Apr 01 2025

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

Cf. A382621.

(* A382621list is defined at A382621 *)
Differences[A382621list[100]]

a(n) = A382621(n+1) - A382621(n).

A382625 Split A382621 into runs of increasing elements. a(n) is the length of the n-th run.

Original entry on oeis.org

2, 2, 2, 2, 12, 2, 13, 2, 2, 7, 6, 2, 9, 2, 2, 5, 4, 7, 8, 9, 10, 11, 4, 4, 3, 2, 4, 3, 4, 2, 6, 2, 4, 7, 2, 2, 11, 2, 2, 39, 2, 5, 5, 2, 5, 5, 2, 5, 6, 2, 2, 9, 2, 2, 7, 2, 5, 7, 2, 2, 2, 11, 2, 2, 2, 2, 10, 2, 6, 4, 6, 2, 7, 4, 7, 2, 8, 4, 8, 2, 9, 2, 11, 2, 2, 5, 2

Offset: 1

Paolo Xausa, Apr 01 2025

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

Cf. A382621.

(* A382621list is defined at A382621 *)
Map[Length, Most[Split[A382621list[500], Less]]]

A382623 Positive integers that contain an even digit d immediately preceded by a digit <= d.

Original entry on oeis.org

12, 14, 16, 18, 22, 24, 26, 28, 34, 36, 38, 44, 46, 48, 56, 58, 66, 68, 78, 88, 100, 102, 104, 106, 108, 112, 114, 116, 118, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 134, 136, 138, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 156, 158, 160, 161, 162, 163

Offset: 1

Paolo Xausa, Apr 01 2025

Conjecture: these are the numbers missing from A382621.

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

Cf. A347298, A382464, A382621, A382624 (complement).

A382623Q[k_] := MemberQ[Partition[IntegerDigits[k], 2, 1], {i_, j_?EvenQ} /; i <= j];
Select[Range[200], A382623Q]

def ok(n):
    s = str(n)
    return any(s[i+1] >= s[i] for i in range(len(s)-1) if s[i+1] in "02468")
print([k for k in range(1, 164) if ok(k)]) # Michael S. Branicky, Apr 03 2025

A382624 Positive integers such that every even digit except the leftmost is immediately preceded by a larger digit.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 15, 17, 19, 20, 21, 23, 25, 27, 29, 30, 31, 32, 33, 35, 37, 39, 40, 41, 42, 43, 45, 47, 49, 50, 51, 52, 53, 54, 55, 57, 59, 60, 61, 62, 63, 64, 65, 67, 69, 70, 71, 72, 73, 74, 75, 76, 77, 79, 80, 81, 82, 83, 84, 85, 86, 87, 89, 90

Offset: 1

Paolo Xausa, Apr 01 2025

Conjecture: these are the terms of A382621, sorted.

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

Cf. A377912, A382465, A382621, A382623 (complement).

A382624Q[k_] := FreeQ[Partition[IntegerDigits[k], 2, 1], {i_, j_?EvenQ} /; i <= j];
Select[Range[100], A382624Q]

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 "02468")
print([k for k in range(1, 91) if ok(k)]) # Michael S. Branicky, Apr 03 2025

A382462 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, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 19, 31, 21, 23, 33, 34, 35, 36, 37, 38, 39, 51, 24, 53, 41, 25, 55, 56, 57, 58, 59, 71, 26, 73, 43, 45, 61, 27, 75, 63, 46, 77, 78, 79, 91, 28, 93, 47, 81, 29, 95, 65, 67, 83, 48, 97, 85, 68, 99, 111, 49, 112, 69

Offset: 1

Paolo Xausa, Mar 27 2025

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

No term contains the digit 0. - Paolo Xausa, Apr 30 2025

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

Similar sequences: A342042, A342043, A342044, A342045, A382621, A382935, A383059.

Cf. A382463, A382464, A382465, A382466.

A382462list[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++] & [Mod[#, 10]*10]; While[s[fu], fu++]; s[a] = True; a) &, 1, nmax-1]];
A382462list[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 "02468")
def agen(): # generator of terms
    an, seen, s, m = 1, {1}, "1", 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

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

A382622 First differences of A382621.

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A382625 Split A382621 into runs of increasing elements. a(n) is the length of the n-th run.

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A382623 Positive integers that contain an even digit d immediately preceded by a digit <= d.

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A382624 Positive integers such that every even digit except the leftmost is immediately preceded by a larger digit.

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A382462 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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Python