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A336005 a(n) is the number of terms in the mixed binary-ternary representation of n. See Comments.

%I A336005 #26 Jan 28 2025 00:49:32
%S A336005 1,1,1,1,2,1,2,1,1,2,2,2,2,3,2,1,2,1,2,2,2,2,3,2,3,2,1,2,2,2,2,1,2,2,
%T A336005 2,2,3,2,3,2,2,3,3,3,3,4,3,2,3,2,3,3,3,1,2,2,2,2,3,2,3,2,2,1,2,2,2,2,
%U A336005 3,2,3,2,2,3,3,3,3,4,3,2,1,2,2,2,2,3
%N A336005 a(n) is the number of terms in the mixed binary-ternary representation of n. See Comments.
%C A336005 Suppose that B1 and B2 are increasing sequences of positive integers, and let B be the increasing sequence of numbers in the union of B1 and B2.  Every positive integer n has a unique representation given by the greedy algorithm with B1 as base, and likewise for B2 and B. For many n, the number of terms in the B-representation of n is less than the number of terms in the B1-representation, as well as the B2-representation, but not for all n, as in the example 45 = 27 + 18 (ternary) and 45 = 32 + 9 + 4 (mixed).
%H A336005 Michael S. Branicky, <a href="/A336005/b336005.txt">Table of n, a(n) for n = 1..10000</a>
%e A336005 7 = 6 + 1, so a(7) = 2.
%e A336005 45 = 32 + 9 + 4, so a(45) = 3.
%t A336005 z = 20; zz = 100;
%t A336005 b1 = Sort[Table[2^k, {k, 0, z}], Greater];
%t A336005 b2 = Sort[Union[Table[3^k, {k, 0, z}], Table[2*3^k, {k, 0, z}]],
%t A336005    Greater]; b = Sort[Union[b1, b2], Greater];
%t A336005 g1 = Map[{#, DeleteCases[b1  Reap[
%t A336005          FoldList[(Sow[Quotient[#1, #2]]; Mod[#1, #2]) &, #, b1]][[2,
%t A336005          1]], 0]} &, Range[zz]];
%t A336005 m1 = Map[Length[#[[2]]] &, g1];
%t A336005 g2 = Map[{#, DeleteCases[b2 Reap[FoldList[(Sow[Quotient[#1, #2]]; Mod[#1, #2]) &, #, b2]][[2, 1]], 0]} &, Range[zz]];
%t A336005 m2 = Map[Length[#[[2]]] &, g2];
%t A336005 g = Map[{#, DeleteCases[
%t A336005      b Reap[FoldList[(Sow[Quotient[#1, #2]]; Mod[#1, #2]) &, #,
%t A336005          b]][[2, 1]], 0]} &, Range[zz]]
%t A336005 m = Map[Length[#[[2]]] &, g];
%t A336005 m1 (* # terms in binary representation *)
%t A336005 m2 (* # terms in ternary representation *)
%t A336005 m  (* # terms in mixed base representation *)  (* A336005 *)
%o A336005 (Python)
%o A336005 from itertools import count, takewhile
%o A336005 N = 10**6
%o A336005 B1 = list(takewhile(lambda x: x[0] <= N, ((2**i, 2) for i in count(0))))
%o A336005 B21 = list(takewhile(lambda x: x[0] <= N, ((3**i, 3) for i in count(0))))
%o A336005 B22 = list(takewhile(lambda x: x[0] <= N, ((2*3**i, 3) for i in count(0))))
%o A336005 B = sorted(set(B1 + B21 + B22), reverse=True)
%o A336005 def gbt(n, B): # greedy binary-ternary representation
%o A336005     r = []
%o A336005     for t, b in B:
%o A336005         if t <= n:
%o A336005             r.append(t)
%o A336005             n -= t
%o A336005             if n == 0:
%o A336005                 return r
%o A336005 def a(n): return len(gbt(n, B))
%o A336005 print([a(n) for n in range(1, 87)]) # _Michael S. Branicky_, Jan 06 2022
%Y A336005 Cf. A336004, A336006.
%K A336005 nonn,base
%O A336005 1,5
%A A336005 _Clark Kimberling_, Jul 06 2020