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

3, 30, 33, 75, 300, 303, 330, 333, 429, 750, 813, 3000, 3003, 3030, 3125, 3300, 3330, 3333, 4290, 4329, 7500, 7575, 8130, 30000, 30003, 30030, 30300, 30303, 31250, 33000, 33300, 33330, 33333, 42900, 43290, 46875, 75000, 75075, 75750, 76923, 81103, 81300, 300000

Offset: 1

Author

Bernard Schott, Jan 30 2022

Keywords

• nonn
• base

Comments

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

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

Some subsequences:

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

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

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

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

Examples

			As 1/33 = 0.0303030303..., 33 is a term.
As 1/75 = 0.0133333333..., 75 is a term.
As 1/429 = 0.002331002331002331..., 429 is a term.
		

Links

• Index entries for sequences related to decimal expansion of 1/n

Crossrefs

Cf. A000533, A094028, A266385, A333236.

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

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

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

Programs

Mathematica

Select[Range[10^5], Max[RealDigits[1/#][[1]]] == 3 &] (* Amiram Eldar, Jan 30 2022 *)

Python

from fractions import Fraction
from itertools import count, islice
from sympy import n_order, multiplicity
def repeating_decimals_expr(f, digits_only=False):
    """ returns repeating decimals of Fraction f as the string aaa.bbb[ccc].
        returns only digits if digits_only=True.
    """
    a, b = f.as_integer_ratio()
    m2, m5 = multiplicity(2,b), multiplicity(5,b)
    r = max(m2,m5)
    k, m = 10**r, 10**n_order(10,b//2**m2//5**m5)-1
    c = k*a//b
    s = str(c).zfill(r)
    if digits_only:
        return s+str(m*k*a//b-c*m)
    else:
        w = len(s)-r
        return s[:w]+'.'+s[w:]+'['+str(m*k*a//b-c*m)+']'
def A350814_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda m:max(repeating_decimals_expr(Fraction(1,m),digits_only=True)) == '3',count(max(startvalue,1)))
A350814_list = list(islice(A350814_gen(),10)) # Chai Wah Wu, Feb 07 2022

Extensions

More terms from Amiram Eldar, Jan 30 2022