A342042 -id:A342042 - cp's OEIS frontend

A342044 When a digit d is odd, the next digit is > d.

0, 1, 2, 3, 4, 5, 6, 7, 8, 12, 13, 40, 14, 15, 60, 16, 17, 80, 18, 20, 21, 22, 23, 41, 24, 25, 61, 26, 27, 81, 28, 34, 35, 62, 36, 37, 82, 38, 42, 43, 44, 45, 63, 46, 47, 83, 48, 56, 57, 84, 58, 64, 65, 66, 67, 85, 68, 78, 86, 87, 88, 120, 121, 200, 122, 123, 400, 124, 125, 600, 126, 127, 800, 128

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 9 is present in the sequence (as 9, being odd, would block it).

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A342044

Cf. A342042, A342043, A342045, A342046 and A342047 (variations on the same idea).

PARI
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See Links section.
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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
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See Links section.
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A347298 Numbers that contain an even digit d immediately followed by a digit <= d.

Original entry on oeis.org

20, 21, 22, 40, 41, 42, 43, 44, 60, 61, 62, 63, 64, 65, 66, 80, 81, 82, 83, 84, 85, 86, 87, 88, 100, 120, 121, 122, 140, 141, 142, 143, 144, 160, 161, 162, 163, 164, 165, 166, 180, 181, 182, 183, 184, 185, 186, 187, 188, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213, 214, 215

Offset: 1

N. J. A. Sloane, Aug 26 2021

Conjecture: This is precisely the list of numbers missing from A342042.

The conjecture is correct, see A342042 for details. - Sebastian Karlsson, Oct 02 2021

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

Cf. A342042, A377912 (complement).

A347298Q[k_] := MemberQ[Partition[IntegerDigits[k], 2, 1], {i_?EvenQ, j_} /; j <= i];
Select[Range[300], A347298Q] (* Paolo Xausa, Mar 17 2025 *)

def ok(n):
    s = str(n)
    return any(s[i] in "2468" and s[i+1] <= s[i] for i in range(len(s)-1))
print([k for k in range(216) if ok(k)]) # Michael S. Branicky, Nov 28 2024

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

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

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.
```

A377917 Number of n-digit terms in A377912.

Original entry on oeis.org

10, 66, 489, 3631, 26951, 200045, 1484850, 11021410, 81807240, 607220362, 4507138581, 33454573430, 248319075015, 1843166918425, 13681044394077, 101548575900358, 753751904485831, 5594779921615960, 41527672679871145, 308242258385100002, 2287951231622970075, 16982489246315828049

Offset: 1

Sebastian Karlsson and N. J. A. Sloane, Nov 30 2024

Also number of n-digit terms in A342042.
a(1149) has 1001 digits. - Michael S. Branicky, Nov 30 2024
The terms of A377912 as decimal digit strings are a regular language so can be counted using the transitions in a state machine matching those strings. - Kevin Ryde, Dec 01 2024

			The 66 two-digit terms in A377912 are
  10,11,12,13,14,15,16,17,18,19,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,
  38,39,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,67,68,69,70,71,72,73,74,
  75,76,77,78,79,89,90,91,92,93,94,95,96,97,98,99.
There is an obvious division into 5 blocks of size 10 and blocks of sizes 7, 5, 3, and 1.

Michael S. Branicky, Table of n, a(n) for n = 1..1148
Michael S. Branicky, Python program for A377917
Index entries for linear recurrences with constant coefficients, signature (5,15,20,15,6,1).

Cf. A342042, A377912.

First differences of A377918.

LinearRecurrence[{5, 15, 20, 15, 6, 1}, {10, 66, 489, 3631, 26951, 200045, 1484850}, 25] (* Paolo Xausa, Dec 01 2024 *)

From Kevin Ryde, Dec 01 2024: (Start)
a(n) = 5*a(n-1) + 15*a(n-2) + 20*a(n-3) + 15*a(n-4) + 6*a(n-5) + a(n-6) for n>=8.
G.f.: -1 + x + (1+x)^4 / (1 - 5*x - 15*x^2 - 20*x^3 - 15*x^4 - 6*x^5 - x^6). (End)

a(6) and beyond from Michael S. Branicky, Nov 30 2024

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A342044 When a digit d is odd, 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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A347298 Numbers that contain an even digit d immediately followed by a digit <= 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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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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A342046 When a digit d is prime, the next digit is > d.

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

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