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A350814 Numbers m such that the largest digit in the decimal expansion of 1/m is 3.

%I A350814 #42 Feb 07 2022 21:46:44
%S A350814 3,30,33,75,300,303,330,333,429,750,813,3000,3003,3030,3125,3300,3330,
%T A350814 3333,4290,4329,7500,7575,8130,30000,30003,30030,30300,30303,31250,
%U A350814 33000,33300,33330,33333,42900,43290,46875,75000,75075,75750,76923,81103,81300,300000
%N A350814 Numbers m such that the largest digit in the decimal expansion of 1/m is 3.
%C A350814 If m is a term, 10*m is also a term.
%C A350814 3 is the only prime up to 2.6*10^8 (see comments in A333237).
%C A350814 Some subsequences:
%C A350814 {3, 30, 300, ...} = A093138 \ {1}.
%C A350814 {3, 33, 333, ...} = A002277 \ {0}.
%C A350814 {3, 33, 303, 3003, ...} = 3 * A000533.
%C A350814 {3, 303, 30303, 3030303, ...} = 3 * A094028.
%H A350814 <a href="/index/1#1overn">Index entries for sequences related to decimal expansion of 1/n</a>
%e A350814 As 1/33 = 0.0303030303..., 33 is a term.
%e A350814 As 1/75 = 0.0133333333..., 75 is a term.
%e A350814 As 1/429 = 0.002331002331002331..., 429 is a term.
%t A350814 Select[Range[10^5], Max[RealDigits[1/#][[1]]] == 3 &] (* _Amiram Eldar_, Jan 30 2022 *)
%o A350814 (Python)
%o A350814 from fractions import Fraction
%o A350814 from itertools import count, islice
%o A350814 from sympy import n_order, multiplicity
%o A350814 def repeating_decimals_expr(f, digits_only=False):
%o A350814     """ returns repeating decimals of Fraction f as the string aaa.bbb[ccc].
%o A350814         returns only digits if digits_only=True.
%o A350814     """
%o A350814     a, b = f.as_integer_ratio()
%o A350814     m2, m5 = multiplicity(2,b), multiplicity(5,b)
%o A350814     r = max(m2,m5)
%o A350814     k, m = 10**r, 10**n_order(10,b//2**m2//5**m5)-1
%o A350814     c = k*a//b
%o A350814     s = str(c).zfill(r)
%o A350814     if digits_only:
%o A350814         return s+str(m*k*a//b-c*m)
%o A350814     else:
%o A350814         w = len(s)-r
%o A350814         return s[:w]+'.'+s[w:]+'['+str(m*k*a//b-c*m)+']'
%o A350814 def A350814_gen(startvalue=1): # generator of terms >= startvalue
%o A350814     return filter(lambda m:max(repeating_decimals_expr(Fraction(1,m),digits_only=True)) == '3',count(max(startvalue,1)))
%o A350814 A350814_list = list(islice(A350814_gen(),10)) # _Chai Wah Wu_, Feb 07 2022
%Y A350814 Cf. A000533, A094028, A266385, A333236.
%Y A350814 Similar with largest digit k: A333402 (k=1), A341383 (k=2), A333237 (k=9).
%Y A350814 Subsequences: A002277 \ {0}, A093138 \ {1}.
%Y A350814 Decimal expansion: A010701 (1/3), A010674 (1/33).
%K A350814 nonn,base
%O A350814 1,1
%A A350814 _Bernard Schott_, Jan 30 2022
%E A350814 More terms from _Amiram Eldar_, Jan 30 2022