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

1,1

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}.

Links

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

Formula

A352153(a(n)) = 0.

Example

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).

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 &]

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.

Sequence in context: A210757 A031979 A055239 * A138708 A061116 A333237

Adjacent sequences: A352151 A352152 A352153 * A352155 A352156 A352157

Keyword

nonn,base

Author

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

Status

approved