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A342266 Lexicographically earliest sequence of distinct nonnegative terms such that both a(n) and a(n) * a(n+1) have digits in nondecreasing order.

%I A342266 #38 Mar 14 2021 11:14:40
%S A342266 0,1,2,3,4,6,8,7,5,9,13,12,14,16,18,26,44,27,17,15,23,29,46,28,48,47,
%T A342266 24,19,117,38,36,33,34,37,67,35,127,114,39,57,78,146,236,58,77,1444,
%U A342266 177,157,2477,144,247,45,25,49,227,147,257,1777,12568,116,68,66,118,113,59,226,148,166,134,167,334
%N A342266 Lexicographically earliest sequence of distinct nonnegative terms such that both a(n) and a(n) * a(n+1) have digits in nondecreasing order.
%C A342266 10 is obviously the first integer not present in the sequence as 1 > 0; 11 will never show either because the result of a(n) * 11 is already in the sequence or because the said result has digits in contradiction with the definition.
%C A342266 It would be good to have a proof that there is an infinite sequence with the desired property. It could happen then any choice for any number of initial terms will eventually fail. - _David A. Corneth_ and _N. J. A. Sloane_, Mar 07 2021
%C A342266 The authors agree, but are unable to give the desired proof. So it is indeed possible that this sequence is wrong from the first term on.
%H A342266 Carole Dubois, <a href="/A342266/a342266_1.txt">Table of n, a(n) for n = 1..138</a> (terms calculated with a(n+1) < a(n)*1000, (backtracking when not found inside this limit)).
%e A342266 a(5) = 4 and a(6) = 6 have product 24: the three numbers have digits in nondecreasing order;
%e A342266 a(6) = 6 and a(7) = 8 have product 48: the three numbers have digits in nondecreasing order;
%e A342266 a(7) = 8 and a(7) = 7 have product 56: the three numbers have digits in nondecreasing order; etc.
%Y A342266 Cf. A009994 (numbers with digits in nondecreasing order), A342264 and A342265 (variations on the same idea).
%K A342266 base,nonn
%O A342266 1,3
%A A342266 _Eric Angelini_ and _Carole Dubois_, Mar 07 2021