A376090 Positive numbers k such that the decimal expansions of k and 1/k have distinct nonzero digits.

2, 4, 5, 8, 9, 11, 12, 15, 18, 20, 22, 25, 36, 39, 40, 44, 45, 50, 55, 66, 72, 74, 75, 80, 88, 90, 99, 101, 108, 110, 111, 120, 125, 132, 135, 148, 150, 180, 182, 198, 200, 202, 205, 220, 222, 225, 239, 250, 259, 264, 271, 275, 296, 351, 360, 369, 375, 378

Offset

1,1

Comments

A digit d > 0 appearing in the decimal expansion of a term does not appear in the decimal expansion of its reciprocal and vice versa.

This sequence is infinite: if m is a term, then 10*m is also a term.

Links

Rémy Sigrist, Table of n, a(n) for n = 1..3000

Rémy Sigrist, Logarithmic scatterplot of the first differences of the first 3000 terms

Rémy Sigrist, C++ program

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

Example

The first terms, alongside their reciprocal (with the repeating part between parentheses), are:
  n   a(n)  1/a(n)
  --  ----  ----------
   1     2  0.5(0)
   2     4  0.25(0)
   3     5  0.2(0)
   4     8  0.125(0)
   5     9  0.(1)
   6    11  0.(09)
   7    12  0.08(3)
   8    15  0.0(6)
   9    18  0.0(5)
  10    20  0.05(0)
  11    22  0.0(45)
  12    25  0.04(0)
  13    36  0.02(7)
  14    39  0.(025641)
  15    40  0.025(0)

Prog

(C++) // See Links section.
(Python)
from itertools import count, islice
from sympy import multiplicity, n_order
def A376090_gen(startvalue=1): # generator of terms >= startvalue
    for a in count(max(startvalue, 1)):
        m2, m5 = (~a & a-1).bit_length(), multiplicity(5, a)
        b = min(m2, m5)
        a2 = a//10**b
        m2 -= b
        m5 -= b
        r = max(m2, m5)
        k, m = 10**r, 10**(t := n_order(10, (a2>>m2)//5**m5))-1
        c = k//a2
        if not len((set(d for d in str(c) if d!='0')|set(d for d in str(m*k//a2-c*m) if d!='0'))&set(d for d in str(a2) if d!='0')):
            yield a
A376090_list = list(islice(A376090_gen(), 30)) # Chai Wah Wu, Sep 14 2024

Crossrefs

Cf. A376089.

Sequence in context: A201818 A047380 A288214 * A117121 A101925 A101884

Adjacent sequences: A376087 A376088 A376089 * A376091 A376092 A376093

Keyword

nonn,base

Author

Rémy Sigrist, Sep 09 2024

Status

approved