cp's OEIS Frontend

A258482 Positive numbers n with concatenations n=x//y such that n=x^2-y^2.

%I A258482 #26 Mar 23 2020 06:24:44
%S A258482 100,147,10000,13467,1000000,1010100,1016127,1034187,1140399,1190475,
%T A258482 1216512,1300624,1334667,1416767,1484847,1530900,100000000,102341547,
%U A258482 102661652,116604399,133346667,159809775,10000000000,10101010100,13333466667,14848484847
%N A258482 Positive numbers n with concatenations n=x//y such that n=x^2-y^2.
%C A258482 The terms in this sequence have only an odd number of digits. If they would have an even number of digits both parts would have the same length. The maximum difference x^2 - y^2 would be (10^m-1)^2 - 1^2, which is (10^m-2)*10^m. But this is always less than (10^m-1)^2 + 1, so m never equals x^2 - y^2.
%C A258482 For example m=3: 999^2 - 1^2 < 999001.
%C A258482 The terms in this sequence all start with the digit '1'. Suppose they would start with the digit '2' (or more) the smallest possiblity of x^2 - y^2 would be (2*10^m)^2 - (10^m-1)^2 = 3*10^2*m + 2*10^m-1, but this is always more than 2*10^2*m + 10^3-1, so m never equals x^2 - y^2.
%C A258482 For example m=3: 2000^2 - 999^2 > 2000999.
%C A258482 This sequence has an infinite subsequence, since (10^m+(10^m+2)/3)*10^m+(2*10^m+1)/3 equals (10^m+(10^m+2)/3)^2 - ((2*10^m+1)/3)^2 for every positive m.
%C A258482 For example m=3: 1334667 = 1334^2 - 667^2.
%C A258482 This set is a subset of A113797.
%H A258482 Giovanni Resta, <a href="/A258482/b258482.txt">Table of n, a(n) for n = 1..4397</a> (terms < 10^60)
%F A258482 n=x*10^d+y, where 10^(d-1)<=x<10^d and 0<=y<10^d and n=x^2-y^2.
%e A258482 147 is a member, since 147 = 14^2 - 7^2.
%e A258482 1484847 is a member, since 1484847 = 1484^2- 847^2.
%e A258482 48 is a member of A113797 since 48 = |4^2 - 8^2|, but 48 is not equal to 4^2 - 8^2, so 48 is not a member of this sequence.
%o A258482 (Python)
%o A258482 for p in range(1, 7):
%o A258482     for i in range(10**p, 10**(p + 1)):
%o A258482         c = 10**(int((p - 1) / 2) + 1)
%o A258482         a, b = i // c, i % c
%o A258482         if i == a**2 - b**2:
%o A258482             print(i, end=",")
%o A258482 (PARI) isok(n) = {d = digits(n); if (#d > 1, for (k=1, #d-1, vba = Vecrev(vector(k, i, d[i])); vbb = Vecrev(vector(#d-k, i, d[k+i])); da = sum(i=1, #vba, vba[i]*10^(i-1)); db = sum(i=1, #vbb, vbb[i]*10^(i-1)); if (da^2 - db^2 == n, return(1));););} \\ _Michel Marcus_, Jun 14 2015
%Y A258482 Cf. A002654, A055616, A064942, A064943, A113797, A101311, A162700, A257796.
%K A258482 nonn,base
%O A258482 1,1
%A A258482 _Pieter Post_, May 31 2015
%E A258482 More terms from _Giovanni Resta_, Jun 14 2015