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A342932 The unique sequence {a(1), a(2), a(3), a(4), ...} of digits 1, 2, or 3 such that the number a(n)a(n-1)...a(2)a(1), read in base 6, is divisible by 3^n.

%I A342932 #48 May 25 2021 12:29:28
%S A342932 3,1,2,3,3,1,1,3,1,2,2,1,1,3,1,3,1,3,3,3,2,3,3,2,1,1,2,1,3,1,2,1,1,3,
%T A342932 3,2,2,3,1,3,2,2,2,3,3,1,2,2,2,2,2,1,3,2,2,3,2,2,1,2,1,3,2,2,3,1,1,1,
%U A342932 1,1,3,2,2,2,3,3,2,1,3,1,1,2,2,3,1,3,2,3,2,3,1,1,3,1,2,3,3,2,3,2,3,1,1,2,3
%N A342932 The unique sequence {a(1), a(2), a(3), a(4), ...} of digits 1, 2, or 3 such that the number a(n)a(n-1)...a(2)a(1), read in base 6, is divisible by 3^n.
%C A342932 The distribution seems to be uniform but random (empirical observation).
%C A342932 To prove that such a digit sequence exists and is unique is a good (but uncommon) example of a proof by induction.
%H A342932 Rémy Sigrist, <a href="/A342932/a342932.png">Colored scatterplot of (n, 3*o(n)-n) for = 1..1000000</a> (where o corresponds to the ordinal transform of the sequence)
%H A342932 Eugen Ionascu, <a href="/A342932/a342932.nb">Mathematica File</a>
%e A342932 3 is divisible by 3^1;
%e A342932 13_6 = 1*6 + 3 = 9, which is divisible by 3^2,
%e A342932 213_6 = 2*6^2 + 1*6 + 3 = 81, which is divisible by 3^3.
%t A342932 nd[n_] := Module[{k, i, s, ss, L, a}, L = Array[f, n]; f[1] = 3;
%t A342932   Do[s = Sum[6^(k - 1)*f[k], {k, 1, i - 1}];
%t A342932    ss = Mod[2^(i - 1)*s/3^(i - 1), 3];
%t A342932    If[ss == 0, f[i] = 3, If[ss == 1, f[i] = 2, f[i] = 1]], {i, 2, n}];
%t A342932   s = Sum[6^(k - 1)*f[k], {k, 1, n}];
%t A342932   {L, s/3^n}]
%o A342932 (Python)
%o A342932 n, div, divnum = 0, 1, 0
%o A342932 while n < 87:
%o A342932     div, a = 3*div, 1
%o A342932     while (a*6**n+divnum)%div != 0:
%o A342932         a = a+1
%o A342932     divnum, n = divnum+a*6**n, n+1
%o A342932     print(a, end=', ') # _A.H.M. Smeets_, Apr 13 2021
%o A342932 (PARI) { q=0; t=1; for (n=1, 105, print1 (d=[3,1,2][1+lift(-q/Mod(t,3))]", "); q=(t*d+q)/3; t*=2) } \\ _Rémy Sigrist_, Apr 15 2021
%Y A342932 Cf. A023396, A053312, A126933.
%K A342932 nonn,base
%O A342932 1,1
%A A342932 _Eugen Ionascu_, Mar 29 2021