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A336004 Numbers whose mixed binary-ternary representation is not a binary or ternary representation. See Comments.

%I A336004 #31 Feb 18 2022 21:06:00
%S A336004 13,14,22,23,26,31,35,38,39,41,42,43,44,45,46,47,50,51,52,53,58,59,62,
%T A336004 67,70,71,73,74,75,76,77,78,79,85,86,89,94,95,97,98,103,104,107,112,
%U A336004 113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,131
%N A336004 Numbers whose mixed binary-ternary representation is not a binary or ternary representation. See Comments.
%C A336004 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).
%C A336004 Note that 1 and 2 = 10_2 = 2_3 are each representable as terms in both binary and ternary. - _Michael S. Branicky_, Jan 06 2022
%e A336004 7 =  6 + 1 = 21_3 is not a term;
%e A336004 11 =  9 + 2 = 102_3 is not a term;
%e A336004 13 =  9 + 4 = 3^2 + 2^2 is a term;
%e A336004 22 = 18 + 4 = 2*3^2 + 2^2 is a term.
%t A336004 u = Table[2^n, {n, 0, 50}]; v = Table[3^n, {n, 0, 40}];
%t A336004 uQ[n_] := MemberQ[u, n]; vQ[n_] := MemberQ[v, n];
%t A336004 Attributes[uQ] = {Listable}; Attributes[vQ] = {Listable};
%t A336004 s = Reverse[Union[Flatten[Table[{2^(n - 1), 3^n}, {n, 1, 30}]]]];
%t A336004 w = Map[#[[1]] &,Select[Map[{#[[1]], {Apply[And, uQ[#[[2]]]],
%t A336004        Apply[And, vQ[#[[2]]]]}} &, Map[{#, DeleteCases[
%t A336004         s Reap[FoldList[(Sow[Quotient[#1, #2]]; Mod[#1, #2]) &, #,
%t A336004             s]][[2, 1]], 0]} &,
%t A336004      Range[700]]], #[[2]] == {False, False} &]]
%t A336004 (* _Peter J. C. Moses_, Jun 14 2020 *)
%o A336004 (Python)
%o A336004 from itertools import count, takewhile
%o A336004 N = 10**6
%o A336004 B1 = list(takewhile(lambda x: x[0] <= N, ((2**i, 2) for i in count(0))))
%o A336004 B21 = list(takewhile(lambda x: x[0] <= N, ((3**i, 3) for i in count(0))))
%o A336004 B22 = list(takewhile(lambda x: x[0] <= N, ((2*3**i, 3) for i in count(0))))
%o A336004 B = sorted(set(B1 + B21 + B22), reverse=True)
%o A336004 def ok(n):
%o A336004     r, bases = [], set()
%o A336004     for t, b in B:
%o A336004         if t <= n:
%o A336004             r.append(t)
%o A336004             if t != 1 and t != 2:
%o A336004                 bases.add(b)
%o A336004             n -= t
%o A336004             if n == 0:
%o A336004                 return len(bases) == 2
%o A336004 print([k for k in range(1, 132) if ok(k)]) # _Michael S. Branicky_, Jan 06 2022
%Y A336004 Cf. A000079, A000244, A336005, A336006, A336007.
%K A336004 nonn,base
%O A336004 1,1
%A A336004 _Clark Kimberling_, Jul 06 2020
%E A336004 Terms and examples corrected by _Michael S. Branicky_, Jan 06 2022