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

50, 200, 338, 450, 578, 800, 1352, 1682, 1800, 2312, 2450, 2738, 3042, 3200, 3362, 4050, 5202, 5408, 5618, 6050, 6728, 7200, 7442, 9248, 9800, 10658, 10952, 12168, 12800, 13448, 15138, 15842, 16200, 16562, 18050, 18818, 20402, 20808, 21632, 22050, 22472, 23762, 24200, 24642, 25538

Offset: 1

Author

Altug Alkan, May 19 2016

Keywords

• nonn

Comments

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,

k^2 = (a^2 + b^2)*(c^2 + d^2) = (a*c + b*d)^2 + (a*d - b*c)^2,

k^2 = (a^2 + b^2)*(c^2 + d^2) = (a*d + b*c)^2 + (a*c - b*d)^2,

k^2 = (a^2 + b^2)*(a^2 + b^2) = (a^2 - b^2)^2 + (2*a*b)^2,

k^2 = (c^2 + d^2)*(c^2 + d^2) = (c^2 - d^2)^2 + (2*c*d)^2.

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.

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

Examples

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

Crossrefs

Cf. A025285, A084645.

Programs

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