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

Author

Bernard Schott and Robert G. Wilson v, Mar 25 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: 2, 18, 132, 148, 14544, ...

Examples

			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.
		

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).

Programs

Mathematica

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

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

Formula

A352153(a(n)) = 5.

Extensions

More terms from Jinyuan Wang, Mar 27 2022