cp's OEIS Frontend

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.

%I A288781 #20 Jun 22 2025 15:18:12
%S A288781 10,18,100,144,154,198,1000,1296,1702,1998,10000,12222,12727,14949,
%T A288781 15049,17271,17776,19998,100000,104878,117343,122221,177777,182655,
%U A288781 195120,199998,1000000,1005291,1038961,1142856,1148148,1181818,1187109,1208494,1318681
%N 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.
%C A288781 The numbers appear to be in pairs that add up to 299...998; e.g., 144 + 154 = 298, 12222 + 17776 = 29998.
%H A288781 Giovanni Resta, <a href="/A288781/b288781.txt">Table of n, a(n) for n = 1..10000</a>
%t A288781 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 *)
%o A288781 (Python)
%o A288781 from math import sqrt
%o A288781 def is_square(n):
%o A288781   root = int(sqrt(n))
%o A288781   return root*root == n
%o A288781 def find_sols(length):
%o A288781     count = 0
%o A288781     k=10**length
%o A288781     for n in range(k,4*k-2):
%o A288781         discr= (2*k-1)*(2*k-1) - 4*(k*(k-1)-(k-1)*n)
%o A288781         if is_square(discr):
%o A288781             count+=1
%o A288781             b=(-(2*k-1)+sqrt(discr))/2
%o A288781             print(n, k+b, n/(k+b))
%o A288781     return count
%o A288781 for i in range(8):
%o A288781     print(find_sols(i))
%Y A288781 Cf. A285273, A288782 (numerators).
%K A288781 nonn,base
%O A288781 1,1
%A A288781 _James Kilfiger_, Jun 15 2017
%E A288781 Definition corrected by and more terms from _Giovanni Resta_, Jun 30 2017