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A337357 "Choix de Collatz": a(n) is the least number of steps required to reach 1 starting from n under substring substitutions of the form k -> T(k) (where T is the Collatz map, A006370).

%I A337357 #13 Aug 24 2020 23:55:15
%S A337357 0,1,7,2,5,8,9,3,11,6,7,8,9,9,7,4,7,10,8,7,7,8,10,9,8,5,7,10,8,8,7,5,
%T A337357 11,6,9,11,6,7,8,8,7,8,11,9,9,6,8,10,8,9,8,6,11,7,9,9,9,8,11,9,7,6,11,
%U A337357 6,9,12,7,7,9,10,8,7,10,7,10,8,10,8,10,9,8
%N A337357 "Choix de Collatz": a(n) is the least number of steps required to reach 1 starting from n under substring substitutions of the form k -> T(k) (where T is the Collatz map, A006370).
%C A337357 This sequence is a variant of "Choix de Bruxelles" (where we consider substring substitutions of the form k <-> 2*k, see A323286):
%C A337357 - we map a positive number n to any number that can be obtained as follows:
%C A337357 - take a nonempty substring s (without leading zero) in the decimal representation of n,
%C A337357 - if the value of s corresponds to an even number, replace s by s/2,
%C A337357 - otherwise replace s by 3*s + 1.
%C A337357 The sequence is well defined:
%C A337357 - the proof is similar to that described in A337321,
%C A337357 - the initial paths to consider here are the following:
%C A337357      1
%C A337357      2 ->  1
%C A337357      3 -> 10 ->  5 -> 16 ->  8 ->  4 ->  2 ->  1
%C A337357      4 ->  2 ->  1
%C A337357      5 -> 16 ->  8 ->  4 ->  2 ->  1
%C A337357      6 ->  3 -> 10 ->  5 -> 16 ->  8 ->  4 ->  2 -> 1
%C A337357      7 -> 22 -> 11 -> 34 -> 32 -> 16 ->  8 ->  4 -> 2 -> 1
%C A337357      8 ->  4 ->  2 ->  1
%C A337357      9 -> 28 -> 24 -> 22 -> 21 -> 64 -> 32 -> 16 -> 8 -> 4 -> 2 -> 1
%C A337357     10 ->  5 -> 16 ->  8 ->  4 ->  2 ->  1
%C A337357     11 -> 34 -> 32 -> 16 ->  8 ->  4 ->  2 ->  1
%H A337357 Rémy Sigrist, <a href="/A337357/b337357.txt">Table of n, a(n) for n = 1..3652</a>
%H A337357 Rémy Sigrist, <a href="/A337357/a337357.gp.txt">PARI program for A337357</a>
%F A337357 a(n) <= A006577(n) (when A006577(n) >= 0).
%o A337357 (PARI) See Links section.
%Y A337357 Cf. A006370, A006577, A323286, A323454, A337321.
%K A337357 nonn,base
%O A337357 1,3
%A A337357 _Rémy Sigrist_ and _N. J. A. Sloane_, Aug 24 2020