A362710 Numbers m such that the decimal expansion of 1/m contains no digit 0, ignoring leading and trailing 0's.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 18, 20, 22, 24, 25, 26, 28, 30, 32, 35, 36, 40, 44, 45, 50, 54, 55, 56, 60, 64, 65, 66, 70, 72, 74, 75, 80, 82, 88, 90, 100, 104, 108, 112, 120, 125, 128, 132, 140, 144, 148, 150, 160, 175, 176, 180, 200, 216, 220, 224, 225, 240, 250, 252, 260, 264

Offset

1,2

Comments

If k is a term, then so is 10*k.

Links

Robert Israel, Table of n, a(n) for n = 1..1000

Example

a(12) = 14 is a term because 1/14 = 0.0714285714... contains no digit 0 except for leading 0's.

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);
  not(member(0, removeInitial0(NonRepeatingPart(q))) or member(0, RepeatingPart(q)))
end proc:
select(filter, [$1..300]);

Mathematica

Select[Range[500], FreeQ[First[RealDigits[1/#]], 0] &] (* Paolo Xausa, Apr 22 2024 *)

Prog

(Python)
from itertools import count, islice
from sympy import multiplicity, n_order
def A362710_gen(startvalue=1): # generator of terms >= startvalue
    for a in count(max(startvalue, 1)):
        m2, m5 = (~a&a-1).bit_length(), multiplicity(5, a)
        k, m = 10**max(m2, m5), 10**(t:=n_order(10, a//(1<<m2)//5**m5))-1
        if not('0' in str(c:=k//a).lstrip('0') or ((w:=str(m*k//a-c*m).zfill(t)) != '0' and '0' in w)):
            yield a
A362710_list = list(islice(A362710_gen(), 30)) # Chai Wah Wu, May 04 2023

Crossrefs

Complement of A352154. Cf. A362579.

Sequence in context: A178862 A178860 A178858 * A243356 A347933 A117296

Adjacent sequences: A362707 A362708 A362709 * A362711 A362712 A362713

Keyword

nonn,base

Author

Robert Israel, Apr 30 2023

Status

approved