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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.

%I A383250 #15 Apr 29 2025 13:24:33
%S A383250 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,
%T A383250 46,47,48,49,50,52,53,54,60,62,63,64,65,66,67,68,69,70,72,73,74,75,76,
%U A383250 80,82,83,84,85,86,87,88,89,90,92,93,94,95,96,97,98,100,102,103
%N 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.
%C A383250 Conjecture: these are the terms of A342045, sorted.
%C A383250 From _Quinn Savitt_, Apr 29 2025: (Start)
%C A383250 Theorem: These are the terms of A342045, sorted.
%C A383250 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.
%C A383250 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)
%H A383250 Paolo Xausa, <a href="/A383250/b383250.txt">Table of n, a(n) for n = 1..10000</a>
%H A383250 Quinn Savitt, <a href="https://www.overleaf.com/read/xztfqhxspjcr#67e2ba">Formal proof that A383250 = sorted(A342045)</a>
%t A383250 A383250Q[k_] := Last[#] != 1 && FreeQ[Partition[#, 2, 1], {i_?OddQ, j_} /; j >= i] & [IntegerDigits[k]];
%t A383250 Select[Range[0, 200], A383250Q]
%o A383250 (Python)
%o A383250 def ok(n):
%o A383250     if n%10 == 1: return False
%o A383250     s = str(n)
%o A383250     return all(d in "02468" or s[i]<d for i, d in enumerate(s, 1) if i < len(s))
%o A383250 print([k for k in range(104) if ok(k)]) # _Michael S. Branicky_, Apr 28 2025
%Y A383250 Cf. A342045, A383249 (complement).
%Y A383250 Cf. A377912, A382465, A382624, A383062, A382938, A383246, A383248.
%K A383250 nonn,base,easy
%O A383250 1,2
%A A383250 _Paolo Xausa_, Apr 26 2025