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

1,3

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

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

Formula

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

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

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

Example

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)

Prog

(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

Crossrefs

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

Cf. A003592, A007732, A132740.

Sequence in context: A213735 A056858 A137251 * A158359 A046716 A371967

Adjacent sequences: A370754 A370755 A370756 * A370758 A370759 A370760

Keyword

nonn,base

Author

Rémy Sigrist, Feb 29 2024

Status

approved