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

1,1

Comments

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

Links

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

Index entries for sequences related to decimal expansion of 1/n.

Example

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).

Maple

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

Mathematica

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

Prog

(Python)
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<<m2)//5**m5)-1
        if '5' in str(c:=k//a) or '5' in str(m*k//a-c*m):
            yield a
A353441_list = list(islice(A353441_gen(), 20)) # Chai Wah Wu, May 01 2023

Crossrefs

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.

Sequence in context: A336463 A358457 A306235 * A004616 A322386 A309583

Adjacent sequences: A353438 A353439 A353440 * A353442 A353443 A353444

Keyword

nonn,base

Author

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

Status

approved