A342044 -id:A342044 - cp's OEIS frontend

A342042 When a digit d in the digit-stream of this sequence is even, the next digit is > d.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 30, 13, 14, 50, 15, 16, 70, 17, 18, 90, 19, 23, 24, 51, 25, 26, 71, 27, 28, 91, 29, 31, 32, 33, 34, 52, 35, 36, 72, 37, 38, 92, 39, 45, 46, 73, 47, 48, 93, 49, 53, 54, 55, 56, 74, 57, 58, 94, 59, 67, 68, 95, 69, 75, 76, 77, 78, 96, 79, 89, 97, 98, 99, 101

Offset: 1

Eric Angelini, Feb 26 2021

The definition refers to the digit-stream in the sequence (ignoring the commas), which starts 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 0, 1, 1, 1, 2, 3, 0, ...
The sequence is always extended with the smallest nonnegative integer not yet present that doesn't lead to a contradiction.
Theorem: The sequence contains every nonnegative integer except those in A347298.
Proved in September 2021. See S.K. link for a new, more detailed proof. - Sebastian Karlsson, Nov 28 2024. See N.J.A.S. link for an alternative, shorter, proof. - N. J. A. Sloane, Nov 29 2024
Comments added by N. J. A. Sloane, Dec 04 2024 (Start):
Let S = present sequence, P = A377912. By definition the terms in P appear in their natural order. There are A377917(k) terms in P of decimal length k >= 1. They form a consecutive block in P, starting at P(i1) and ending at P(i2), where i1 = A377918(k), i2 = A377918(k+1)-1.
We know S contains exactly the same terms as P, but in a different order.
Conjecture 1. For k >= 1, the terms of length k in S form a consecutive block with the same starting and ending points as in P. In both P and S, the block begins with 10101... (1's and 0's alternate, length is k) and end with 99...9 (k 9's).
Conjecture 2. We know every number appears in S. Suppose x = S(m) = 899...9 (with k-1 9's). Then x is the last term of length k in S that begins with a digit <= 8. The remaining terms of length k have leading digit 9 and appear in order, ending with 99...9 (k 9's).
(Some k-digit numbers beginning with 9 may appear before x.)
(End)
Comment from N. J. A. Sloane, Dec 01 2024 (Start)
Let c1 = 7.422574840... and c2 = 1.3824387... be the constants defined in A377918. Then assuming Conjecture 1, the index of the last term of length k in the present sequence is close to (c2*c1^k, 10^k). [Thanks to Sebastian Karlsson for pointing out that Conjecture 1 is required and is as yet unproved.]
  Let x = c2*c1^k, and express k in terms of x.
Then this point has coordinates (x,y) where y = (x/c2)^c3, with c3 = (log 10)/(log c1) = 1.14869... This defines a curve that is a good approximation to the lower envelope of the present sequence.
For example, the fifth meeting point has coordinates (31148, 101010) (see A377918) and the formula here gives (x,y) = (31148, 100003.0039).
(End)
Comment from Sebastian Karlsson, Dec 12 2024: (Start)
Theorem: Let d be in {1, 2, ..., 8}. For every positive integer k, the k-digit number d99...9 appears in the sequence before the k-digit number (d+1)99...9.
A proof can be found in the links. Since all k-digit numbers starting with 9 appears before any (k+1)-digit number, we get that terms of a certain length form a consecutive block. In particular, this proves Conjectures 1 and 2 above.
(End)

N. J. A. Sloane, Table of n, a(n) for n = 1..20000 [Computed using Rémy Sigrist's PARI program. The first 10000 terms were computed by Peter Kagey]
Sebastian Karlsson, Proof of Theorem.
Sebastian Karlsson, Proof of the second Theorem.
Rémy Sigrist, Colored scatterplot of the first 20000 terms (where the color is function of the leading digit of a(n))
Rémy Sigrist, PARI program for A342042
N. J. A. Sloane, A Nasty Surprise in a Sequence and Other OEIS Stories, Experimental Mathematics Seminar, Rutgers University, Oct 10 2024, Youtube video; Slides [Mentions this sequence]
N. J. A. Sloane, An Alternative Proof of the Theorem.

Cf. A342043, A342044, A342045, A342046 and A342047 (variations on the same idea).
See A377913 and A377914 for records.
See also A347298.
Cf. A379901, A379903.

PARI
```
\\ See Links section.
```

def cond(s, minfirst):
    return all(s[i+1] > s[i] for i in range(len(s)-1) if s[i] in "02468")
def aupton(terms):
    alst, seen = [0], {0}
    while len(alst) < terms:
        d = alst[-1]%10
        an = minfirst = (1 - d%2)*(d+1)
        stran = str(an)
        while an in seen or not cond(stran, minfirst):
            an += 1
            stran = str(an)
            if int(stran[0]) < minfirst:
                an = minfirst*10**(len(stran)-1)
        alst.append(an); seen.add(an)
    return alst
print(aupton(77)) # Michael S. Branicky, Sep 07 2021

Edited by N. J. A. Sloane, Nov 24 2024

A342043 When a digit d is even, the next digit is < d.

Original entry on oeis.org

1, 2, 11, 3, 4, 12, 13, 5, 6, 14, 15, 7, 8, 16, 17, 9, 18, 19, 21, 31, 32, 111, 33, 34, 35, 36, 37, 38, 39, 41, 42, 112, 113, 43, 51, 52, 114, 115, 53, 54, 116, 55, 56, 57, 58, 59, 61, 62, 117, 63, 64, 118, 65, 71, 72, 119, 73, 74, 121, 75, 76, 131, 77, 78, 79, 81, 82, 132, 133, 83, 84, 134, 135, 85

Offset: 1

Eric Angelini, Feb 26 2021

The sequence is always extended with the smallest positive integer not yet present that doesn't lead to a contradiction.

No digit 0 is present in the sequence (as 0, being even, would block it).

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A342043

Cf. A342042, A342044, A342045, A342046 and A342047 (variations on the same idea).

PARI
```
See Links section.
```

A342045 When a digit d is odd, the next digit is < d.

Original entry on oeis.org

0, 2, 3, 10, 4, 5, 20, 6, 7, 22, 8, 9, 23, 24, 25, 26, 27, 28, 29, 30, 32, 40, 42, 43, 100, 44, 45, 46, 47, 48, 49, 50, 52, 53, 102, 54, 60, 62, 63, 103, 104, 64, 65, 105, 106, 66, 67, 68, 69, 70, 72, 73, 107, 108, 74, 75, 109, 76, 80, 82, 83, 200, 84, 85, 202, 86, 87, 203, 204, 88, 89, 205, 206, 90, 92, 93

Offset: 1

Eric Angelini, Feb 26 2021

After a(1) = 0, the sequence is always extended with the smallest positive integer not yet present that doesn't lead to a contradiction.

No term ends in 1 (as this 1 would block the sequence).

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A342045

Cf. A342042, A342043, A342044, A342046 and A342047 (variations on the same idea).

PARI
```
See Links section.
```

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

A383247 Positive integers that contain the digit 9, or an odd digit d immediately followed by a digit <= d.

Original entry on oeis.org

9, 10, 11, 19, 29, 30, 31, 32, 33, 39, 49, 50, 51, 52, 53, 54, 55, 59, 69, 70, 71, 72, 73, 74, 75, 76, 77, 79, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 129, 130, 131, 132, 133

Offset: 1

Paolo Xausa, Apr 25 2025

Conjecture: these are the numbers missing from A342044.

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

Cf. A342044, A383248 (complement).

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

A383247Q[k_] := MemberQ[#, 9] || MemberQ[Partition[#, 2, 1], {i_?OddQ, j_} /; j <= i] & [IntegerDigits[k]];
Select[Range[200], A383247Q]

def ok(n):
    s = str(n)
    return "9" in s or any(d in "13579" and s[i]<=d for i, d in enumerate(s, 1) if i < len(s))
print([k for k in range(134) if ok(k)]) # 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

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

A342046 When a digit d is prime, the next digit is > d.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 34, 13, 40, 14, 15, 60, 16, 17, 80, 18, 19, 23, 41, 24, 25, 61, 26, 27, 81, 28, 29, 35, 62, 36, 37, 82, 38, 39, 42, 43, 44, 45, 63, 46, 47, 83, 48, 49, 56, 57, 84, 58, 59, 64, 65, 66, 67, 85, 68, 69, 78, 79, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99

Offset: 1

Eric Angelini, Feb 26 2021

After a(1) = 0, the sequence is always extended with the smallest positive integer not yet present that doesn't lead to a contradiction.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A342046

Cf. A342042, A342043, A342044, A342045 and A342047 (variations on the same idea).

PARI
```
See Links section.
```

A342047 When a digit d is prime, the next digit is < d.

Original entry on oeis.org

0, 1, 2, 10, 3, 11, 4, 5, 12, 13, 14, 6, 7, 15, 16, 8, 9, 17, 18, 19, 20, 21, 30, 31, 32, 100, 40, 41, 42, 101, 43, 102, 103, 104, 44, 45, 46, 47, 48, 49, 50, 51, 52, 105, 106, 53, 107, 54, 60, 61, 62, 108, 63, 109, 64, 65, 110, 66, 67, 68, 69, 70, 71, 72, 111, 73, 112, 113, 114, 74, 75, 115, 116

Offset: 1

Eric Angelini, Feb 26 2021

After a(1) = 0, the sequence is always extended with the smallest positive integer not yet present that doesn't lead to a contradiction.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A342047

Cf. A342042, A342043, A342044, A342045 and A342046 (variations on the same idea).

PARI
```
See Links section.
```

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A342042 When a digit d in the digit-stream of this sequence is even, the next digit is > d.

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A342043 When a digit d is even, the next digit is < d.

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A342045 When a digit d is odd, the next digit is < d.

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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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A383247 Positive integers that contain the digit 9, or an odd digit d immediately followed by a digit <= d.

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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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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