A342932 - OEIS

%I #48 May 25 2021 12:29:28

%S 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 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 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 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 The distribution seems to be uniform but random (empirical observation).

%C To prove that such a digit sequence exists and is unique is a good (but uncommon) example of a proof by induction.

%H 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 Eugen Ionascu, <a href="/A342932/a342932.nb">Mathematica File</a>

%e 3 is divisible by 3^1;

%e 13_6 = 1*6 + 3 = 9, which is divisible by 3^2,

%e 213_6 = 2*6^2 + 1*6 + 3 = 81, which is divisible by 3^3.

%t nd[n_] := Module[{k, i, s, ss, L, a}, L = Array[f, n]; f[1] = 3;

%t   Do[s = Sum[6^(k - 1)*f[k], {k, 1, i - 1}];

%t    ss = Mod[2^(i - 1)*s/3^(i - 1), 3];

%t    If[ss == 0, f[i] = 3, If[ss == 1, f[i] = 2, f[i] = 1]], {i, 2, n}];

%t   s = Sum[6^(k - 1)*f[k], {k, 1, n}];

%t   {L, s/3^n}]

%o (Python)

%o n, div, divnum = 0, 1, 0

%o while n < 87:

%o     div, a = 3*div, 1

%o     while (a*6**n+divnum)%div != 0:

%o         a = a+1

%o     divnum, n = divnum+a*6**n, n+1

%o     print(a, end=', ') # _A.H.M. Smeets_, Apr 13 2021

%o (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 Cf. A023396, A053312, A126933.

%K nonn,base

%O 1,1

%A _Eugen Ionascu_, Mar 29 2021