%I #23 Apr 05 2022 09:40:24

%S 2,18,20,132,148,180,200,1320,1480,1800,2000,13008,13200,14544,14800,

%T 18000,20000,130080,132000,145440,148000,180000,200000,1300800,

%U 1320000,1454400,1480000,1734375,1800000,2000000,11521152,12890625,13008000,13200000,14544000,14800000

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

%C Leading 0's are not considered, otherwise every integer >= 11 would be a term (see examples).

%C 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 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, ...

%F A352153(a(n)) = 5.

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

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

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

%o (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

%o (Python)

%o from itertools import count,islice

%o from sympy import multiplicity,n_order

%o def A352159_gen(startvalue=1): # generator of terms >= startvalue

%o for n in count(max(startvalue,1)):

%o m2, m5 = multiplicity(2,n), multiplicity(5,n)

%o k, m = 10**max(m2,m5), 10**(t := n_order(10,n//2**m2//5**m5))-1

%o c = k//n

%o s = str(m*k//n-c*m).zfill(t)

%o if s == '0' and min(str(c)) == '5':

%o yield n

%o elif '0' not in s and min(str(c).lstrip('0')+s) == '5':

%o yield n

%o A352159_list = list(islice(A352159_gen(),10)) # _Chai Wah Wu_, Mar 28 2022

%Y Cf. A351471.

%Y Subsequence: A093136 \ {0}.

%Y 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).

%K nonn,base

%O 1,1

%A _Bernard Schott_ and _Robert G. Wilson v_, Mar 25 2022

%E More terms from _Jinyuan Wang_, Mar 27 2022