A350814 - OEIS

%I #42 Feb 07 2022 21:46:44

%S 3,30,33,75,300,303,330,333,429,750,813,3000,3003,3030,3125,3300,3330,

%T 3333,4290,4329,7500,7575,8130,30000,30003,30030,30300,30303,31250,

%U 33000,33300,33330,33333,42900,43290,46875,75000,75075,75750,76923,81103,81300,300000

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

%C If m is a term, 10*m is also a term.

%C 3 is the only prime up to 2.6*10^8 (see comments in A333237).

%C Some subsequences:

%C {3, 30, 300, ...} = A093138 \ {1}.

%C {3, 33, 333, ...} = A002277 \ {0}.

%C {3, 33, 303, 3003, ...} = 3 * A000533.

%C {3, 303, 30303, 3030303, ...} = 3 * A094028.

%H <a href="/index/1#1overn">Index entries for sequences related to decimal expansion of 1/n</a>

%e As 1/33 = 0.0303030303..., 33 is a term.

%e As 1/75 = 0.0133333333..., 75 is a term.

%e As 1/429 = 0.002331002331002331..., 429 is a term.

%t Select[Range[10^5], Max[RealDigits[1/#][[1]]] == 3 &] (* _Amiram Eldar_, Jan 30 2022 *)

%o (Python)

%o from fractions import Fraction

%o from itertools import count, islice

%o from sympy import n_order, multiplicity

%o def repeating_decimals_expr(f, digits_only=False):

%o     """ returns repeating decimals of Fraction f as the string aaa.bbb[ccc].

%o         returns only digits if digits_only=True.

%o     """

%o     a, b = f.as_integer_ratio()

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

%o     r = max(m2,m5)

%o     k, m = 10**r, 10**n_order(10,b//2**m2//5**m5)-1

%o     c = k*a//b

%o     s = str(c).zfill(r)

%o     if digits_only:

%o         return s+str(m*k*a//b-c*m)

%o     else:

%o         w = len(s)-r

%o         return s[:w]+'.'+s[w:]+'['+str(m*k*a//b-c*m)+']'

%o def A350814_gen(startvalue=1): # generator of terms >= startvalue

%o     return filter(lambda m:max(repeating_decimals_expr(Fraction(1,m),digits_only=True)) == '3',count(max(startvalue,1)))

%o A350814_list = list(islice(A350814_gen(),10)) # _Chai Wah Wu_, Feb 07 2022

%Y Cf. A000533, A094028, A266385, A333236.

%Y Similar with largest digit k: A333402 (k=1), A341383 (k=2), A333237 (k=9).

%Y Subsequences: A002277 \ {0}, A093138 \ {1}.

%Y Decimal expansion: A010701 (1/3), A010674 (1/33).

%K nonn,base

%O 1,1

%A _Bernard Schott_, Jan 30 2022

%E More terms from _Amiram Eldar_, Jan 30 2022