A362579 -id:A362579 - cp's OEIS frontend

A353441 Integers m such that the decimal expansion of 1/m contains the digit 5.

2, 4, 7, 8, 14, 16, 17, 18, 19, 20, 22, 23, 26, 28, 29, 31, 32, 34, 35, 38, 39, 40, 42, 43, 46, 47, 49, 51, 53, 54, 56, 57, 58, 59, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 74, 76, 78, 79, 80, 81, 82, 83, 85, 86, 87, 89, 92, 93, 94, 95, 97, 98, 102, 103, 104, 105, 106, 107, 108, 109

Offset: 1

Bernard Schott and Robert G. Wilson v, Apr 25 2022

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

			m = 7 is a term since 1/7 = 0.142857142857...
m = 22 is a term since 1/22 = 0.04545454545... (here, 5 is the largest digit).
m = 132 is a term since 1/693 = 0.00757575... (here, 5 is the smallest digit).

Robert Israel, Table of n, a(n) for n = 1..10000
Index entries for sequences related to decimal expansion of 1/n.

A351471 (largest digit=5) and A352159 (smallest digit=5) are subsequences.
Similar with digit k: A352154 (k=0), A353437 (k=1), A353438 (k=2), A353439 (k=3), A353440 (k=4), this sequence (k=5), A353442 (k=6), A353443 (k=7), A353444 (k=8), A333237 (k=9).
Complement of A362579.

filter:= proc(n) local q;
  q:= NumberTheory:-RepeatingDecimal(1/n);
  member(5,RepeatingPart(q)) or member(5, NonRepeatingPart(q))
end proc:
select(filter, [$1..200]); # Robert Israel, Apr 25 2023

f[n_] := Union[ Flatten[ RealDigits[ 1/n][[1]] ]]; Select[ Range@ 125, MemberQ[f@#, 5] &]

from itertools import count, islice
from sympy import multiplicity, n_order
def A353441_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**n_order(10,a//(1<A353441_list = list(islice(A353441_gen(),20)) # Chai Wah Wu, May 01 2023

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

Original entry on oeis.org

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

Robert Israel, Apr 30 2023

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

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

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

Complement of A352154. Cf. A362579.

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]);

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

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<A362710_list = list(islice(A362710_gen(),30)) # Chai Wah Wu, May 04 2023

cp's OEIS Frontend

A353441 Integers m such that the decimal expansion of 1/m contains the digit 5.

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