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

Author

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

Keywords

• nonn
• base

Comments

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

Examples

			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).
		

Links

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

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.

Programs

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] &]

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<A353441_list = list(islice(A353441_gen(),20)) # Chai Wah Wu, May 01 2023