A352159 Numbers m such that the smallest digit in the decimal expansion of 1/m is 5, ignoring leading and trailing 0's.

2, 18, 20, 132, 148, 180, 200, 1320, 1480, 1800, 2000, 13008, 13200, 14544, 14800, 18000, 20000, 130080, 132000, 145440, 148000, 180000, 200000, 1300800, 1320000, 1454400, 1480000, 1734375, 1800000, 2000000, 11521152, 12890625, 13008000, 13200000, 14544000, 14800000

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: 2, 18, 132, 148, 14544, ...

Links

Table of n, a(n) for n=1..36.

Formula

A352153(a(n)) = 5.

Example

m = 148 is a term since 1/148 = 0.00675675675... and the smallest digit after the leading 0's is 5.
m = 1320 is a term since 1/1320 = 0.000075757575... and the smallest digit after the leading 0's is 5.

Mathematica

f[n_] := Union[ Flatten[ RealDigits[ 1/n][[1]] ]]; Select[ Range@ 1100, Min@ f@# == 5 &]

Prog

(PARI) is(n) = my(d=#digits(n-1), m=9, r=10^d, x=valuation(n, 2), y=valuation(n, 5)); for(k=1, max(x, y)-d*(n==x=2^x*5^y)+znorder(Mod(10, n/x)), if(5>m=min(m, r\n), return(0)); r=r%n*10); m==5; \\ Jinyuan Wang, Mar 27 2022
(Python)
from itertools import count, islice
from sympy import multiplicity, n_order
def A352159_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)) == '5':
            yield n
        elif '0' not in s and min(str(c).lstrip('0')+s) == '5':
                yield n
A352159_list = list(islice(A352159_gen(), 10)) # Chai Wah Wu, Mar 28 2022

Crossrefs

Cf. A351471.

Subsequence: A093136 \ {0}.

Similar with smallest digit k: A352154 (k=0), A352155 (k=1), A352156 (k=2), A352157 (k=3), A352158 (k=4), this sequence (k=5), A352160 (k=6), A352153 (no known term for k=7), A352161 (k=8), no term (k=9).

Sequence in context: A066242 A022371 A299380 * A266956 A092587 A247457

Adjacent sequences: A352156 A352157 A352158 * A352160 A352161 A352162

Keyword

nonn,base

Author

Bernard Schott and Robert G. Wilson v, Mar 25 2022

Extensions

More terms from Jinyuan Wang, Mar 27 2022

Status

approved