A352160 Numbers m such that the smallest digit in the decimal expansion of 1/m is k = 6, ignoring leading and trailing 0's.

15, 150, 1500, 15000, 103125, 150000, 1031250, 1500000, 10312500, 15000000, 103125000, 130078125, 150000000, 1031250000, 1300781250, 1500000000, 10312500000

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 from 1 (A003592) would be terms.

If t is a term, 10*t is also a term; so, terms with no trailing zeros are all primitive terms: 15, 103125, 130078125, ...

Note that for k = 7, if any term exists, it must be greater than 10^10. - Jinyuan Wang, Mar 28 2022

Links

Table of n, a(n) for n=1..17.

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

Formula

A352153(a(n)) = 6.

Example

m = 150 is a term since 1/150 = 0.0066666666... and the smallest digit after the leading 0's is 6.
m = 103125 is a term since 1/103125 = 0.000009696969... and the smallest digit after the leading 0's is 6.

Mathematica

f[n_] := Union[ Flatten[ RealDigits[ 1/n][[1]] ]]; Select[ Range@ 1100, Min@ f@# == 6 &]

Prog

(Python)
from itertools import count, islice
from sympy import multiplicity, n_order
def A352160_gen(startvalue=1): # generator of terms >= startvalue
    for n in count(max(startvalue, 1)):
        m2, m5 = multiplicity(2, n), multiplicity(5, n)
        k, m = 10**max(m2, m5), 10**(t := n_order(10, n//2**m2//5**m5))-1
        c = k//n
        s = str(m*k//n-c*m).zfill(t)
        if '0' not in s and min(str(c).lstrip('0')+s) == '6':
            yield n
A352160_list = list(islice(A352160_gen(), 5)) # Chai Wah Wu, Mar 28 2022

Crossrefs

Cf. A351472.

Similar with smallest digit k: A352154 (k=0), A352155 (k=1), A352156 (k=2), A352157 (k=3), A352158 (k=4), A352159 (k=5), this sequence (k=6), A352153 (no known term for k=7), A352161 (k=8), no term (k=9).

Sequence in context: A081135 A084902 A021364 * A323298 A206366 A016103

Adjacent sequences: A352157 A352158 A352159 * A352161 A352162 A352163

Keyword

nonn,base,more

Author

Bernard Schott and Robert G. Wilson v, Mar 27 2022

Extensions

a(9)-a(17) from Jinyuan Wang, Mar 28 2022

Status

approved