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

Author

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

Keywords

• nonn
• base
• more

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

Examples

			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.
		

Links

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

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).

Programs

Mathematica

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

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

Formula

A352153(a(n)) = 6.

Extensions

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