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
Links
- Giovanni Resta, Table of n, a(n) for n = 1..10000
Programs
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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 *) -
Python
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))
Extensions
Definition corrected by and more terms from Giovanni Resta, Jun 30 2017
Comments