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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.

%I A288782 #24 Jun 22 2025 15:18:04
%S A288782 10,34,100,208,238,394,1000,1680,2898,3994,10000,14938,16198,22348,
%T A288782 22648,29830,31600,39994,100000,109994,137694,149380,316048,333630,
%U A288782 380720,399994,1000000,1010610,1079440,1306120,1318244,1396694,1409228,1460458,1738920,1768810,1826150
%N 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.
%C A288782 The values 399..994 all seem to appear.
%H A288782 Giovanni Resta, <a href="/A288782/b288782.txt">Table of n, a(n) for n = 1..10000</a>
%t A288782 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 *)
%o A288782 (Python)
%o A288782 from math import sqrt
%o A288782 def is_square(n):
%o A288782   root = int(sqrt(n))
%o A288782   return root*root == n
%o A288782 def find_sols(length):
%o A288782     count = 0
%o A288782     k=10**length
%o A288782     for n in range(k,4*k-2):
%o A288782         discr= (2*k-1)*(2*k-1) - 4*(k*(k-1)-(k-1)*n)
%o A288782         if is_square(discr):
%o A288782             count+=1
%o A288782             b=(-(2*k-1)+sqrt(discr))/2
%o A288782             print(n, k+b, n/(k+b))
%o A288782     return count
%o A288782 for i in range(8):
%o A288782     print(find_sols(i))
%Y A288782 Cf. A285273, A288781 (denominators).
%K A288782 nonn,base
%O A288782 1,1
%A A288782 _James Kilfiger_, Jun 15 2017
%E A288782 Definition corrected by _Giovanni Resta_, Jun 30 2017