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A360190 Starting from 1, successively take the smallest "Choix de Bruxelles" with factor 13 which is not already in the sequence.

%I A360190 #43 Jan 09 2025 13:03:49
%S A360190 1,13,133,1333,13333,133333,1333333,125641,1256413,12564133,1197241,
%T A360190 117481,9037,90391,9031,90313,903133,90241,902413,9024133,90241333,
%U A360190 6941641,693241,6932413,69324133,6717241,671557,65557,5557,55591,5431,54313,543133,54241
%N A360190 Starting from 1, successively take the smallest "Choix de Bruxelles" with factor 13 which is not already in the sequence.
%C A360190 At a given term t, the Choix de Bruxelles with factor 13 can choose to multiply any decimal digit substring (not starting 0) of t by 13, or divide by 13 if that substring is divisible by 13.
%C A360190 These choices on substrings give various possible next values and here take the smallest not yet in the sequence.
%C A360190 The sequence is finite and ends at a(6851) = 7, since the sole next Choix there is multiplication by 13 to 91, but 91 is already in the sequence at the preceding a(6850) = 91.
%H A360190 Alon Vinkler, <a href="/A360190/b360190.txt">Table of n, a(n) for n = 0..6851</a>
%H A360190 Eric Angelini, Lars Blomberg, Charlie Neder, Remy Sigrist, and N. J. A. Sloane, <a href="http://arxiv.org/abs/1902.01444">"Choix de Bruxelles": A New Operation on Positive Integers</a>, arXiv:1902.01444 [math.NT], Feb 2019; Fib. Quart. 57:3 (2019), 195-200.
%H A360190 Eric Angelini, Lars Blomberg, Charlie Neder, Remy Sigrist, and N. J. A. Sloane,, <a href="/A307635/a307635.pdf">"Choix de Bruxelles": A New Operation on Positive Integers</a>, Local copy.
%H A360190 Alon Vinkler, <a href="/A360190/a360190_2.txt">C# Program</a>
%e A360190 Below, square brackets [] represent multiplication by 13 (e.g., [4] = 52); curly brackets {} represent division by 13 (e.g., {26} = 2); digits outside the brackets are not affected by the multiplication or division (e.g., 1[3] = 139 and 1{169} = 113).
%e A360190 We begin with 1 and, at each step, we go to the smallest number possible that hasn't yet appeared in the sequence:
%e A360190  1 --> [1] = 13
%e A360190  13 --> [1]3 = 133
%e A360190  133 --> [1]33 = 1333
%e A360190  1333 --> [1]333 = 13333
%e A360190  13333 --> [1]3333 = 133333
%e A360190  133333 --> [1]33333 = 1333333
%e A360190  1333333 --> 1{333333} = 125641
%e A360190  ... and so on.
%o A360190 (C#) // See Links
%Y A360190 Cf. A358708 (steps by factor 2), A323286 (Choix with factor 2).
%K A360190 nonn,base,fini,full
%O A360190 0,2
%A A360190 _Alon Vinkler_, Jan 29 2023