OFFSET
1,1
COMMENTS
The distribution seems to be uniform but random (empirical observation).
To prove that such a digit sequence exists and is unique is a good (but uncommon) example of a proof by induction.
LINKS
Rémy Sigrist, Colored scatterplot of (n, 3*o(n)-n) for = 1..1000000 (where o corresponds to the ordinal transform of the sequence)
Eugen Ionascu, Mathematica File
EXAMPLE
3 is divisible by 3^1;
13_6 = 1*6 + 3 = 9, which is divisible by 3^2,
213_6 = 2*6^2 + 1*6 + 3 = 81, which is divisible by 3^3.
MATHEMATICA
nd[n_] := Module[{k, i, s, ss, L, a}, L = Array[f, n]; f[1] = 3;
Do[s = Sum[6^(k - 1)*f[k], {k, 1, i - 1}];
ss = Mod[2^(i - 1)*s/3^(i - 1), 3];
If[ss == 0, f[i] = 3, If[ss == 1, f[i] = 2, f[i] = 1]], {i, 2, n}];
s = Sum[6^(k - 1)*f[k], {k, 1, n}];
{L, s/3^n}]
PROG
(Python)
n, div, divnum = 0, 1, 0
while n < 87:
div, a = 3*div, 1
while (a*6**n+divnum)%div != 0:
a = a+1
divnum, n = divnum+a*6**n, n+1
print(a, end=', ') # A.H.M. Smeets, Apr 13 2021
(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
CROSSREFS
KEYWORD
nonn,base
AUTHOR
Eugen Ionascu, Mar 29 2021
STATUS
approved