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Numbers m such that the decimal expansion of 1/m contains the digit 0, ignoring leading and trailing 0's.
17

%I #29 Jul 01 2023 16:53:53

%S 11,13,17,19,21,23,27,29,31,33,34,37,38,39,41,42,43,46,47,48,49,51,52,

%T 53,57,58,59,61,62,63,67,68,69,71,73,76,77,78,79,81,83,84,85,86,87,89,

%U 91,92,93,94,95,96,97,98,99,101,102,103,105,106,107,109,110,111,113,114

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

%C Leading 0's are not considered, otherwise every integer >= 11 would be a term (see examples).

%C 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 If k is a term, 10*k is also a term; so, terms with no trailing zeros are all primitive.

%C Some subsequences:

%C {11, 111, 1111, ...} = A002275 \ {0, 1}

%C {33, 333, 3333, ...} = A002277 \ {0, 3}.

%C {77, 777, 7777, ...} = A002281 \ {0, 7}

%C {11, 101, 1001, 10001, ...} = A000533 \ {1}.

%H Robert Israel, <a href="/A352154/b352154.txt">Table of n, a(n) for n = 1..10000</a>

%F A352153(a(n)) = 0.

%e m = 13 is a term since 1/13 = 0.0769230769230769230... has a periodic part = '07692307' or '76923070' with a 0.

%e 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 removeInitial0:= proc(L) local i;

%p for i from 1 to nops(L) do if L[i] <> 0 then return L[i..-1] fi od;

%p []

%p end proc:

%p filter:= proc(n) local q;

%p q:= NumberTheory:-RepeatingDecimal(1/n);

%p member(0, removeInitial0(NonRepeatingPart(q))) or member(0, RepeatingPart(q))

%p end proc:

%p select(filter, [$1..300]); # _Robert Israel_, Apr 26 2023

%t f[n_] := Union[ Flatten[ RealDigits[ 1/n][[1]] ]]; Select[ Range@ 200, Min@ f@# == 0 &]

%Y Cf. A333402, A341383, A350814.

%Y 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 Cf. A000533, A002275, A002277, A002281.

%K nonn,base

%O 1,1

%A _Bernard Schott_ and _Robert G. Wilson v_, Mar 14 2022