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

1,1

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 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, ...

Links

Table of n, a(n) for n=1..41.

Formula

A352153(a(n)) = 3.

Example

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.

Mathematica

f[n_] := Union[ Flatten[ RealDigits[ 1/n][[1]] ]]; Select[ Range@ 1100, Min@ f@# == 3 &]

Prog

(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

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).

Sequence in context: A073952 A107231 A363913 * A293656 A131936 A009135

Adjacent sequences: A352154 A352155 A352156 * A352158 A352159 A352160

Keyword

nonn,base

Author

Bernard Schott and Robert G. Wilson v, Mar 19 2022

Status

approved