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A273293 Numbers n such that n and n^2 are the sums of two nonzero squares in exactly two ways. 0

%I

%S 50,200,338,450,578,800,1352,1682,1800,2312,2450,2738,3042,3200,3362,

%T 4050,5202,5408,5618,6050,6728,7200,7442,9248,9800,10658,10952,12168,

%U 12800,13448,15138,15842,16200,16562,18050,18818,20402,20808,21632,22050,22472,23762,24200,24642,25538

%N Numbers n such that n and n^2 are the sums of two nonzero squares in exactly two ways.

%C If n is the sum of 2 nonzero squares in exactly 2 ways, then n = a^2 + b^2 = c^2 + d^2 where (a, b), (c, d) are distinct and a, b, c, d are nonzero. For n^2;

%C n^2 = (a^2 + b^2)*(c^2 + d^2) = (a*c + b*d)^2 + (a*d - b*c)^2,

%C n^2 = (a^2 + b^2)*(c^2 + d^2) = (a*d + b*c)^2 + (a*c - b*d)^2,

%C n^2 = (a^2 + b^2)*(a^2 + b^2) = (a^2 - b^2)^2 + (2*a*b)^2,

%C n^2 = (c^2 + d^2)*(c^2 + d^2) = (c^2 - d^2)^2 + (2*c*d)^2.

%C Note that if n is of the form 2*m^2 where m is nonzero integer, then the first two representations will be the same and one of the last two identities will not be the sum of two nonzero squares and we will have two distinct representations for n^2. This is the case that gives motivation for this sequence.

%e 50 is a term because 50 = 1^2 + 7^2 = 5^2 + 5^2 and 2500 = 14^2 + 48^2 = 30^2 + 40^2.

%o (PARI) isA273293(n) = {nb = 0; lim = sqrtint(n); for (x=1, lim, if ((n-x^2 >= x^2) && issquare(n-x^2), nb++); ); nb == 2; }

%o lista(nn) = for(n=1, nn, if(isA273293(n) && isA273293(n^2), print1(n, ", ")));

%Y Cf. A025285.

%K nonn

%O 1,1

%A _Altug Alkan_, May 19 2016

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Last modified October 20 20:18 EDT 2019. Contains 328273 sequences. (Running on oeis4.)