A370757 a(n) is the least k > 0 such that 1/n and 1/k have equivalent repeating decimal digits.

1, 1, 3, 1, 1, 6, 7, 1, 9, 1, 11, 3, 13, 7, 6, 1, 17, 18, 19, 1, 21, 22, 23, 6, 1, 26, 27, 7, 29, 3, 31, 1, 33, 17, 7, 36, 37, 19, 39, 1, 41, 42, 43, 44, 45, 23, 47, 3, 49, 1, 51, 13, 53, 54, 55, 7, 57, 29, 59, 6, 61, 31, 63, 1, 26, 66, 67, 17, 69, 7, 71, 72

Offset: 1

Author

Rémy Sigrist, Feb 29 2024

Keywords

• nonn
• base

Comments

In other words, a(n) is the least k > 0 such that the fractional parts of (10^i)/n and (10^j)/k are equal for some integers i, j.

a(n) is not always a divisor of n. For example, a(65) = 26 is not a divisor of 65. - David A. Corneth, Mar 01 2024

Examples

			The first terms, alongside the decimal expansion of 1/n with its repeating decimal digits in parentheses, are:
  n   a(n)  1/n
  --  ----  -----------
   1     1  1.(0)
   2     1  0.5(0)
   3     3  0.(3)
   4     1  0.25(0)
   5     1  0.2(0)
   6     6  0.1(6)
   7     7  0.(142857)
   8     1  0.125(0)
   9     9  0.(1)
  10     1  0.1(0)
  11    11  0.(09)
  12     3  0.08(3)
  13    13  0.(076923)
  14     7  0.07(142857)
  15     6  0.0(6)
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..10000
• Rémy Sigrist, PARI program
• Index entries for sequences related to decimal expansion of 1/n

Crossrefs

Cf. A000265 (base-2 analog), A038502 (base-3 analog), A132739 (base-5 analog), A242603 (base-7 analog).

Cf. A003592, A007732, A132740.

Programs

PARI

\\ See Links section.

Python

from itertools import count
from sympy import multiplicity, n_order
def A370757(n):
    m2, m5 = (~n & n-1).bit_length(), multiplicity(5,n)
    r = max(m2,m5)
    w, m = 10**r, 10**(t:=n_order(10,n2) if (n2:=(n>>m2)//5**m5)>1 else 1)-1
    c = w//n
    s = str(m*w//n-c*m).zfill(t)
    l = len(s)
    for k in count(1):
        m2, m5 = (~k & k-1).bit_length(), multiplicity(5,k)
        r = max(m2,m5)
        w, m = 10**r, 10**(t:=n_order(10,k2) if (k2:=(k>>m2)//5**m5)>1 else 1)-1
        c = w//k
        if any(s[i:]+s[:i] == str(m*w//k-c*m).zfill(t) for i in range(l)):
            return k # Chai Wah Wu, Mar 03 2024

Formula

a(n) = 1 iff n belongs to A003592.

a(10*n) = a(n).

A007732(a(n)) = A007732(n).