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

11, 13, 17, 19, 21, 23, 27, 29, 31, 33, 34, 37, 38, 39, 41, 42, 43, 46, 47, 48, 49, 51, 52, 53, 57, 58, 59, 61, 62, 63, 67, 68, 69, 71, 73, 76, 77, 78, 79, 81, 83, 84, 85, 86, 87, 89, 91, 92, 93, 94, 95, 96, 97, 98, 99, 101, 102, 103, 105, 106, 107, 109, 110, 111, 113, 114

Offset: 1

Author

Bernard Schott and Robert G. Wilson v, Mar 14 2022

Keywords

• nonn
• base

Comments

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

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.

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

Some subsequences:

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

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

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

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

Examples

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

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A333402, A341383, A350814.

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).

Cf. A000533, A002275, A002277, A002281.

Programs

Maple

removeInitial0:= proc(L) local i;
  for i from 1 to nops(L) do if L[i] <> 0 then return L[i..-1] fi od;
  []
end proc:
filter:= proc(n) local q;
  q:= NumberTheory:-RepeatingDecimal(1/n);
  member(0, removeInitial0(NonRepeatingPart(q))) or member(0, RepeatingPart(q))
end proc:
select(filter, [$1..300]); # Robert Israel, Apr 26 2023

Mathematica

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

Formula

A352153(a(n)) = 0.