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

1, 13, 133, 1333, 13333, 133333, 1333333, 125641, 1256413, 12564133, 1197241, 117481, 9037, 90391, 9031, 90313, 903133, 90241, 902413, 9024133, 90241333, 6941641, 693241, 6932413, 69324133, 6717241, 671557, 65557, 5557, 55591, 5431, 54313, 543133, 54241

Offset: 0

Author

Alon Vinkler, Jan 29 2023

Keywords

• nonn
• base
• fini
• full

Comments

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.

These choices on substrings give various possible next values and here take the smallest not yet in the sequence.

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.

Examples

			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).
We begin with 1 and, at each step, we go to the smallest number possible that hasn't yet appeared in the sequence:
 1 --> [1] = 13
 13 --> [1]3 = 133
 133 --> [1]33 = 1333
 1333 --> [1]333 = 13333
 13333 --> [1]3333 = 133333
 133333 --> [1]33333 = 1333333
 1333333 --> 1{333333} = 125641
 ... and so on.
		

Links

• Alon Vinkler, Table of n, a(n) for n = 0..6851
• Eric Angelini, Lars Blomberg, Charlie Neder, Remy Sigrist, and N. J. A. Sloane, "Choix de Bruxelles": A New Operation on Positive Integers, arXiv:1902.01444 [math.NT], Feb 2019; Fib. Quart. 57:3 (2019), 195-200.
• Eric Angelini, Lars Blomberg, Charlie Neder, Remy Sigrist, and N. J. A. Sloane,, "Choix de Bruxelles": A New Operation on Positive Integers, Local copy.
• Alon Vinkler, C# Program

Crossrefs

Cf. A358708 (steps by factor 2), A323286 (Choix with factor 2).