A288782 Integers k that have the property that there exists an integer x with n+1 digits, such that 1 <= k/x < 2 and k/x = 1 + (x-10^n)/(10^n-1), i.e., the same digits appear in the denominator and in the recurring decimal.

10, 34, 100, 208, 238, 394, 1000, 1680, 2898, 3994, 10000, 14938, 16198, 22348, 22648, 29830, 31600, 39994, 100000, 109994, 137694, 149380, 316048, 333630, 380720, 399994, 1000000, 1010610, 1079440, 1306120, 1318244, 1396694, 1409228, 1460458, 1738920, 1768810, 1826150

Offset

1,1

Comments

The values 399..994 all seem to appear.

Links

Giovanni Resta, Table of n, a(n) for n = 1..10000

Mathematica

Union @@ Reap[ Do[Sow[k /. List@ToRules@ Reduce[k/x == 1 + (x - 10^n)/(10^n - 1) &&  10^n <= x < 10^(n + 1) && x <= k < 2 x, {k, x}, Integers]], {n, 6}]][[2, 1]] (* Giovanni Resta, Jun 30 2017 *)

Prog

(Python 3)
from math import sqrt
def is_square(n):
  root = int(sqrt(n))
  return root*root == n
def find_sols(length):
    count = 0
    k=10**length
    for n in range(k, 4*k-2):
        discr= (2*k-1)*(2*k-1) - 4*(k*(k-1)-(k-1)*n)
        if is_square(discr):
            count+=1
            b=(-(2*k-1)+sqrt(discr))/2
            print(n, k+b, n/(k+b))
    return count
for i in range(8):
    print(find_sols(i))

Crossrefs

Cf. A285273, A288781 (denominators).

Sequence in context: A009924 A297721 A019257 * A020877 A119171 A119229

Adjacent sequences: A288779 A288780 A288781 * A288783 A288784 A288785

Keyword

nonn,base

Author

James Kilfiger, Jun 15 2017

Extensions

Definition corrected by Giovanni Resta, Jun 30 2017

Status

approved