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A253802 a(n) gives the odd leg of one of the two Pythagorean triangles with hypotenuse A080109(n) = A002144(n)^2. This is the smaller of the two possible odd legs.

%I A253802 #14 Jan 10 2017 05:01:54
%S A253802 7,65,161,41,1081,369,1241,671,721,3471,959,9401,4681,1695,3281,7599,
%T A253802 10199,24521,3439,18335,37241,45241,24465,29281,64001,18561,31855,
%U A253802 27761,76601,7825
%N A253802 a(n) gives the odd leg of one of the two Pythagorean triangles with hypotenuse A080109(n) = A002144(n)^2. This is the smaller of the two possible odd legs.
%C A253802 The corresponding even legs are given in 4*A253803.
%C A253802 The legs of the other Pythagorean triangle with hypotenuse A080109(n) are given A253804(n) (odd) and A253805(n) (even).
%C A253802 Each fourth power of a prime of the form 1 (mod 4) (see A002144(n)^2 = A080175(n)) has exactly two representations as sum of two positive squares (Fermat). See the Dickson reference, (B) on p. 227.
%C A253802 This means that there are exactly two Pythagorean triangles (modulo leg exchange) for each hypotenuse A080109(n) = A002144(n)^2, n >= 1. See the Dickson reference, (A) on p. 227.
%C A253802 Note that the Pythagorean triangles are not always primitive. E.g., n = 2: (65, 4*39, 13^2) = 13*(5, 4*3, 13). For each prime congruent 1 (mod 4) (A002144) there is one and only one such non-primitive triangle with hypotenuse p^2 (just scale the unique primitive triangle with hypotenuse p with the factor p). Therefore, one of the two existing Pythagorean triangles with hypotenuse from A080109 is primitive and the other is imprimitive.
%D A253802 L. E. Dickson, History of the Theory of Numbers, Carnegie Institution, Publ. No. 256, Vol. II, Washington D.C., 1920, p. 227.
%F A253802 A080175(n) = A002144(n)^4 = a(n)^2 + (4*A253803(n))^2,
%F A253802 n >= 1, that is,
%F A253802 a(n) = sqrt(A080175(n) - (4*A253803(n))^2), n >= 1.
%e A253802 n = 7: A080175(7) = 7890481 = 53^4 = 2809^2; A002144(7)^4  =  a(7)^2 + (4*A253803(7))^2 = 1241^2 + (4*630)^2.
%e A253802 The other Pythagorean triangle with hypotenuse 53^2 = 2809 has odd leg A253804(7) = 2385 and even leg 4*A253305(7) = 4*371 = 1484: 53^4 = 2385^2 + (4*371)^2.
%Y A253802 Cf. A002144, A002972, A002973, A070079, A070151, A080109, A253305, A253803, A253804.
%K A253802 nonn,easy
%O A253802 1,1
%A A253802 _Wolfdieter Lang_, Jan 14 2015