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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. 3
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 (list; graph; refs; listen; history; text; internal format)
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 A171767

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

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Last modified December 4 02:40 EST 2021. Contains 349469 sequences. (Running on oeis4.)