cp's OEIS Frontend

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

%I A273293 #28 Jan 07 2022 13:30:23
%S A273293 50,200,338,450,578,800,1352,1682,1800,2312,2450,2738,3042,3200,3362,
%T A273293 4050,5202,5408,5618,6050,6728,7200,7442,9248,9800,10658,10952,12168,
%U A273293 12800,13448,15138,15842,16200,16562,18050,18818,20402,20808,21632,22050,22472,23762,24200,24642,25538
%N A273293 Numbers k such that k and k^2 are the sums of two nonzero squares in exactly two ways.
%C A273293 If k is the sum of 2 nonzero squares in exactly 2 ways, then k = a^2 + b^2 = c^2 + d^2 where (a, b), (c, d) are distinct and a, b, c, d are nonzero. For k^2,
%C A273293    k^2 = (a^2 + b^2)*(c^2 + d^2) = (a*c + b*d)^2 + (a*d - b*c)^2,
%C A273293    k^2 = (a^2 + b^2)*(c^2 + d^2) = (a*d + b*c)^2 + (a*c - b*d)^2,
%C A273293    k^2 = (a^2 + b^2)*(a^2 + b^2) = (a^2 - b^2)^2 + (2*a*b)^2,
%C A273293    k^2 = (c^2 + d^2)*(c^2 + d^2) = (c^2 - d^2)^2 + (2*c*d)^2.
%C A273293 Note that if k is of the form 2*m^2 where m is a 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 k^2. This is the case that gives motivation for this sequence.
%C A273293 a(n) is the sum of the areas of the squares on the sides of the integer-sided triangle with hypotenuse A084645(n). - _Wesley Ivan Hurt_, Jan 05 2022
%e A273293 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 A273293 (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 A273293 lista(nn) = for(n=1, nn, if(isA273293(n) && isA273293(n^2), print1(n, ", ")));
%Y A273293 Cf. A025285, A084645.
%K A273293 nonn
%O A273293 1,1
%A A273293 _Altug Alkan_, May 19 2016