

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 + (x10^h)/(10^h1), 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
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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*k2):
discr= (2*k1)*(2*k1)  4*(k*(k1)(k1)*n)
if is_square(discr):
count+=1
b=((2*k1)+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



