A352157 Numbers m such that the smallest digit in the decimal expansion of 1/m is 3, ignoring leading and trailing 0's.

3, 12, 30, 120, 264, 275, 296, 300, 1200, 1875, 2112, 2640, 2664, 2750, 2952, 2960, 3000, 10656, 11808, 12000, 18750, 21120, 22944, 26016, 26400, 26640, 27500, 28125, 29088, 29520, 29600, 30000, 103424, 106560, 106656, 118080, 120000, 156288, 187500, 211200, 229440

Offset: 1

Author

Bernard Schott and Robert G. Wilson v, Mar 19 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: 3, 12, 264, 275, 296, 1875, ...

Examples

			m = 12 is a term since 1/12 = 0.08333333... and the smallest term after the leading 0 is 3.
m = 264 is a term since 1/264 = 0.003787878... and the smallest term after the leading 0's is 3.
		

Crossrefs

Cf. A093138 \ {1} (subsequence), A350814.

Similar with smallest digit k: A352154 (k=0), A352155 (k=1), A352156 (k=2), this sequence (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@# == 3 &]

Python

from itertools import count, islice
from sympy import multiplicity, n_order
def A352157_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) == '3':
            yield n
A352157_list = list(islice(A352157_gen(),20)) # Chai Wah Wu, Mar 28 2022

Formula

A352153(a(n)) = 3.