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%I #26 Mar 23 2020 06:24:44
%S 100,147,10000,13467,1000000,1010100,1016127,1034187,1140399,1190475,
%T 1216512,1300624,1334667,1416767,1484847,1530900,100000000,102341547,
%U 102661652,116604399,133346667,159809775,10000000000,10101010100,13333466667,14848484847
%N Positive numbers n with concatenations n=x//y such that n=x^2-y^2.
%C 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 For example m=3: 999^2 - 1^2 < 999001.
%C 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 For example m=3: 2000^2 - 999^2 > 2000999.
%C 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 For example m=3: 1334667 = 1334^2 - 667^2.
%C This set is a subset of A113797.
%H Giovanni Resta, <a href="/A258482/b258482.txt">Table of n, a(n) for n = 1..4397</a> (terms < 10^60)
%F n=x*10^d+y, where 10^(d-1)<=x<10^d and 0<=y<10^d and n=x^2-y^2.
%e 147 is a member, since 147 = 14^2 - 7^2.
%e 1484847 is a member, since 1484847 = 1484^2- 847^2.
%e 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 (Python)
%o for p in range(1, 7):
%o for i in range(10**p, 10**(p + 1)):
%o c = 10**(int((p - 1) / 2) + 1)
%o a, b = i // c, i % c
%o if i == a**2 - b**2:
%o print(i, end=",")
%o (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 Cf. A002654, A055616, A064942, A064943, A113797, A101311, A162700, A257796.
%K nonn,base
%O 1,1
%A _Pieter Post_, May 31 2015
%E More terms from _Giovanni Resta_, Jun 14 2015
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