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A352160 Numbers m such that the smallest digit in the decimal expansion of 1/m is k = 6, ignoring leading and trailing 0's.

%I A352160 #42 Apr 04 2022 09:48:43
%S A352160 15,150,1500,15000,103125,150000,1031250,1500000,10312500,15000000,
%T A352160 103125000,130078125,150000000,1031250000,1300781250,1500000000,
%U A352160 10312500000
%N A352160 Numbers m such that the smallest digit in the decimal expansion of 1/m is k = 6, ignoring leading and trailing 0's.
%C A352160 Leading 0's are not considered, otherwise every integer >= 11 would be a term (see examples).
%C A352160 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.
%C A352160 If t is a term, 10*t is also a term; so, terms with no trailing zeros are all primitive terms: 15, 103125, 130078125, ...
%C A352160 Note that for k = 7, if any term exists, it must be greater than 10^10. - _Jinyuan Wang_, Mar 28 2022
%H A352160 <a href="/index/1#1overn">Index entries for sequences related to decimal expansion of 1/n</a>.
%F A352160 A352153(a(n)) = 6.
%e A352160 m = 150 is a term since 1/150 = 0.0066666666... and the smallest digit after the leading 0's is 6.
%e A352160 m = 103125 is a term since 1/103125 = 0.000009696969... and the smallest digit after the leading 0's is 6.
%t A352160 f[n_] := Union[ Flatten[ RealDigits[ 1/n][[1]] ]]; Select[ Range@ 1100, Min@ f@# == 6 &]
%o A352160 (Python)
%o A352160 from itertools import count, islice
%o A352160 from sympy import multiplicity, n_order
%o A352160 def A352160_gen(startvalue=1): # generator of terms >= startvalue
%o A352160     for n in count(max(startvalue,1)):
%o A352160         m2, m5 = multiplicity(2,n), multiplicity(5,n)
%o A352160         k, m = 10**max(m2,m5), 10**(t := n_order(10,n//2**m2//5**m5))-1
%o A352160         c = k//n
%o A352160         s = str(m*k//n-c*m).zfill(t)
%o A352160         if '0' not in s and min(str(c).lstrip('0')+s) == '6':
%o A352160             yield n
%o A352160 A352160_list = list(islice(A352160_gen(),5)) # _Chai Wah Wu_, Mar 28 2022
%Y A352160 Cf. A351472.
%Y A352160 Similar with smallest digit k: A352154 (k=0), A352155 (k=1), A352156 (k=2), A352157 (k=3), A352158 (k=4), A352159 (k=5), this sequence (k=6), A352153 (no known term for k=7), A352161 (k=8), no term (k=9).
%K A352160 nonn,base,more
%O A352160 1,1
%A A352160 _Bernard Schott_ and _Robert G. Wilson v_, Mar 27 2022
%E A352160 a(9)-a(17) from _Jinyuan Wang_, Mar 28 2022