A383250 Nonnegative integers not ending with the digit 1 and such that every odd digit except the rightmost is immediately followed by a smaller digit.

0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 20, 22, 23, 24, 25, 26, 27, 28, 29, 30, 32, 40, 42, 43, 44, 45, 46, 47, 48, 49, 50, 52, 53, 54, 60, 62, 63, 64, 65, 66, 67, 68, 69, 70, 72, 73, 74, 75, 76, 80, 82, 83, 84, 85, 86, 87, 88, 89, 90, 92, 93, 94, 95, 96, 97, 98, 100, 102, 103

Offset

1,2

Comments

Conjecture: these are the terms of A342045, sorted.

From Quinn Savitt, Apr 29 2025: (Start)

Theorem: These are the terms of A342045, sorted.

The proof formally defines an extendibility condition: a finite set of selected numbers remains extendible if, for every finite subset, there exists a new number satisfying the "odd digit implies next smaller digit" rule and not ending in 1.

Using induction and monotonicity of extendibility, it follows that every number satisfying these conditions eventually appears in the greedy construction of A342045. This implies that the sequence is a sorted version of A342045. (End)

Links

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

Quinn Savitt, Formal proof that A383250 = sorted(A342045)

Mathematica

A383250Q[k_] := Last[#] != 1 && FreeQ[Partition[#, 2, 1], {i_?OddQ, j_} /; j >= i] & [IntegerDigits[k]];
Select[Range[0, 200], A383250Q]

Prog

(Python)
def ok(n):
    if n%10 == 1: return False
    s = str(n)
    return all(d in "02468" or s[i]<d for i, d in enumerate(s, 1) if i < len(s))
print([k for k in range(104) if ok(k)]) # Michael S. Branicky, Apr 28 2025

Crossrefs

Cf. A342045, A383249 (complement).

Cf. A377912, A382465, A382624, A383062, A382938, A383246, A383248.

Sequence in context: A308393 A009995 A190219 * A038367 A214958 A161350

Adjacent sequences: A383247 A383248 A383249 * A383251 A383252 A383253

Keyword

nonn,base,easy

Author

Paolo Xausa, Apr 26 2025

Status

approved