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A352154 Numbers m such that the decimal expansion of 1/m contains the digit 0, ignoring leading and trailing 0's.

%I A352154 #29 Jul 01 2023 16:53:53
%S A352154 11,13,17,19,21,23,27,29,31,33,34,37,38,39,41,42,43,46,47,48,49,51,52,
%T A352154 53,57,58,59,61,62,63,67,68,69,71,73,76,77,78,79,81,83,84,85,86,87,89,
%U A352154 91,92,93,94,95,96,97,98,99,101,102,103,105,106,107,109,110,111,113,114
%N A352154 Numbers m such that the decimal expansion of 1/m contains the digit 0, ignoring leading and trailing 0's.
%C A352154 Leading 0's are not considered, otherwise every integer >= 11 would be a term (see examples).
%C A352154 Trailing 0's are also not considered, otherwise numbers of the form 2^i*5^j with i, j >= 0, apart 1 (A003592) would be terms.
%C A352154 If k is a term, 10*k is also a term; so, terms with no trailing zeros are all primitive.
%C A352154 Some subsequences:
%C A352154 {11, 111, 1111, ...} = A002275 \ {0, 1}
%C A352154 {33, 333, 3333, ...} = A002277 \ {0, 3}.
%C A352154 {77, 777, 7777, ...} = A002281 \ {0, 7}
%C A352154 {11, 101, 1001, 10001, ...} = A000533 \ {1}.
%H A352154 Robert Israel, <a href="/A352154/b352154.txt">Table of n, a(n) for n = 1..10000</a>
%F A352154 A352153(a(n)) = 0.
%e A352154 m = 13 is a term since 1/13 = 0.0769230769230769230... has a periodic part = '07692307' or '76923070' with a 0.
%e A352154 m = 14 is not a term since 1/14 = 0.0714285714285714285... has a periodic part = '714285' which has no 0 (the only 0 is a leading 0).
%p A352154 removeInitial0:= proc(L) local i;
%p A352154   for i from 1 to nops(L) do if L[i] <> 0 then return L[i..-1] fi od;
%p A352154   []
%p A352154 end proc:
%p A352154 filter:= proc(n) local q;
%p A352154   q:= NumberTheory:-RepeatingDecimal(1/n);
%p A352154   member(0, removeInitial0(NonRepeatingPart(q))) or member(0, RepeatingPart(q))
%p A352154 end proc:
%p A352154 select(filter, [$1..300]); # _Robert Israel_, Apr 26 2023
%t A352154 f[n_] := Union[ Flatten[ RealDigits[ 1/n][[1]] ]]; Select[ Range@ 200, Min@ f@# == 0 &]
%Y A352154 Cf. A333402, A341383, A350814.
%Y A352154 Similar with smallest digit k: this sequence (k=0), A352155 (k=1), A352156 (k=2), A352157 (k=3), A352158 (k=4), A352159 (k=5), A352160 (k=6), A352153 (no known term for k=7), A352161 (k=8), no term (k=9).
%Y A352154 Cf. A000533, A002275, A002277, A002281.
%K A352154 nonn,base
%O A352154 1,1
%A A352154 _Bernard Schott_ and _Robert G. Wilson v_, Mar 14 2022