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A375211 Complement of A129459.

%I A375211 #48 Feb 07 2025 09:26:18
%S A375211 3,4,5,7,9,16,18,19,23,25,33,34,38,43,45,48,49,54,56,58,59,66,77,79,
%T A375211 84,86,93,144,149,155,157,166,176,189,195,223,233,238,243,258,266,293,
%U A375211 295,304,308,313,314,323,333,334,344,376,378,388,393,404,423,433,435,443,448,457,459,566,576,579
%N A375211 Complement of A129459.
%C A375211 Given positive integer x, let y be the greatest integer < x that is not in the sequence (and thus is a member of A129459). Then x is in the sequence iff neither x nor y share any decimal digit with x * y.
%C A375211 For each k >= 1, at least one of (10^k - 1)/3 = 3...33 and (10^k + 2)/3 = 3...34 is in the sequence, as their product is (10^(2*k) + 10^k - 2)/9 = 1..12...2.  Similarly, at least one of (2*10^k - 2)/3 = 6...66 and (2*10^k + 1)/3 = 6...67 is in the sequence.  In particular, the sequence is infinite.
%H A375211 Robert Israel, <a href="/A375211/b375211.txt">Table of n, a(n) for n = 1..5000</a>
%e A375211 a(6) = 16 is a term because 15 is the greatest integer < 16 that is not in the sequence, and neither 16 nor 15 shares a digit with 16 * 15 = 240.
%p A375211 f:= proc(n) local Ln, Lk,k;
%p A375211   Ln:= convert(convert(n,base,10),set);
%p A375211   for k from n+1 do
%p A375211      Lk:= convert(convert(k,base,10),set) union Ln;
%p A375211      if convert(convert(n*k,base,10),set) intersect Lk <> {} then return k fi
%p A375211   od
%p A375211 end proc:
%p A375211 R:= NULL: x:= 0: count:= 0:
%p A375211 while count < 100 do
%p A375211   y:= f(x);
%p A375211   count:= count + y - x - 1;
%p A375211   R:= R, $(x+1)..(y-1);
%p A375211   x:= y
%p A375211 od:
%p A375211 R;
%Y A375211 Cf. A129459.
%K A375211 nonn,base,look
%O A375211 1,1
%A A375211 _Robert Israel_, Feb 06 2025