A352155 Numbers m such that the smallest digit in the decimal expansion of 1/m is 1, ignoring leading and trailing 0's.

1, 6, 7, 8, 9, 10, 14, 24, 26, 28, 32, 35, 54, 55, 56, 60, 64, 65, 66, 70, 72, 74, 75, 80, 82, 88, 90, 100, 104, 112, 128, 140, 175, 176, 224, 240, 260, 280, 320, 350, 432, 448, 468, 504, 512, 528, 540, 548, 550, 560, 572, 576, 584, 592, 600, 616, 625, 640, 650, 660

Offset: 1

Author

Bernard Schott and Robert G. Wilson v, Mar 17 2022

Keywords

• nonn
• base

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

{8, 88, 888, ...} = A002282 \ {0} is a subsequence.

Examples

			m = 14 is a term since 1/14 = 0.0714285714285714285... and the smallest term after the leading 0 is 1.
m = 240 is a term since 1/240 = 0.00416666666... and the smallest term after the leading 0's is 1.
m = 888 is a term since 1/888 = 0.001126126126... and the smallest term after the leading 0's is 1.
		

Crossrefs

Cf. A002282, A333402.

Similar with smallest digit k: A352154 (k=0), this sequence (k=1), A352156 (k=2), A352157 (k=3), A352158 (k=4), A352159 (k=5), A352160 (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@# == 1 &]

Python

from itertools import count, islice
from sympy import multiplicity, n_order
def A352155_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 s == '0' and min(str(c)) == '1':
            yield n
        elif '0' not in s and min(str(c).lstrip('0')+s) == '1':
                yield n
A352155_list = list(islice(A352155_gen(),20)) # Chai Wah Wu, Mar 28 2022

Formula

A352153(a(n)) = 1.