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

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Last modified December 8 17:01 EST 2021. Contains 349596 sequences. (Running on oeis4.)