A288781 Integers x with h+1 digits that have the property that there exists an integer k, with x <= k < 2*x, such that k/x = 1 + (x-10^h)/(10^h-1), i.e., the same digits appear in the denominator and in the recurring decimal.

10, 18, 100, 144, 154, 198, 1000, 1296, 1702, 1998, 10000, 12222, 12727, 14949, 15049, 17271, 17776, 19998, 100000, 104878, 117343, 122221, 177777, 182655, 195120, 199998, 1000000, 1005291, 1038961, 1142856, 1148148, 1181818, 1187109, 1208494, 1318681

Offset

1,1

Comments

The numbers appear to be in pairs that add up to 299...998; e.g., 144 + 154 = 298, 12222 + 17776 = 29998.

Links

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

Mathematica

Union @@ Reap[Do[Sow[x /. 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, A288782 (numerators).

Sequence in context: A068642 A198309 A167342 * A233451 A177172 A358985

Adjacent sequences: A288778 A288779 A288780 * A288782 A288783 A288784

Keyword

nonn,base

Author

James Kilfiger, Jun 15 2017

Extensions

Definition corrected by and more terms from Giovanni Resta, Jun 30 2017

Status

approved