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A268312 First number of the periodic part of the "Say what you see" trajectory (see A005151) of n.

%I A268312 #21 Jan 19 2025 11:45:54
%S A268312 1031223314,21322314,21322314,21322314,21322314,3122331415,3122331416,
%T A268312 3122331417,3122331418,3122331419,1031223314,21322314,21322314,
%U A268312 21322314,21322314,3122331415,3122331416,3122331417,3122331418,3122331419,10311233,21322314,22,21322314,31123314,31123315
%N A268312 First number of the periodic part of the "Say what you see" trajectory (see A005151) of n.
%C A268312 a(40) is the first time the periodic part of the trajectory contains more than one term.
%H A268312 Julien Kluge, <a href="/A268312/b268312.txt">Table of n, a(n) for n = 0..10000</a>
%e A268312 Consider the starting value n = 5. We see one five: 15. We have one one and one five: 1115. We have three ones and one five: 3115... We reach 3122331415 which produces itself. So a(5) = 3122331415.
%t A268312 a005151[n_, m_] :=
%t A268312   FromDigits[
%t A268312    Reverse /@
%t A268312      Sort[Tally[
%t A268312        If[n == 2, m, a005151[n - 1, m]] //
%t A268312         IntegerDigits], #1[[1]] < #2[[1]] &] // Flatten];
%t A268312 a[n_] := Block[{previousNum = 0, currentNum = 1, knownNums = {n}},
%t A268312   For[i = 2, currentNum != previousNum, ++i,
%t A268312    previousNum = currentNum;
%t A268312    currentNum = a005151[i, n];
%t A268312    If[MemberQ[knownNums, currentNum], Return[currentNum],
%t A268312     AppendTo[knownNums, currentNum]];
%t A268312    ];
%t A268312   Return[currentNum];
%t A268312   ]
%t A268312 a /@ Range[0, 100]
%Y A268312 A005151 shows a(1) at term number 13.
%Y A268312 Cf. A047841.
%K A268312 nonn,easy,base
%O A268312 0,1
%A A268312 _Julien Kluge_, Jan 31 2016