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Ordered even leg lengths k (listed with multiplicity) of primitive Pythagorean triangles such that all odd prime factors of k are congruent to 1 (mod 4) and at least one prime factor is congruent to 1 (mod 4).
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%I #48 Jun 22 2022 02:29:27

%S 20,20,40,40,52,52,68,68,80,80,100,100,104,104,116,116,136,136,148,

%T 148,160,160,164,164,200,200,208,208,212,212,232,232,244,244,260,260,

%U 260,260,272,272,292,292,296,296,320,320,328,328,340,340,340,340,356,356

%N Ordered even leg lengths k (listed with multiplicity) of primitive Pythagorean triangles such that all odd prime factors of k are congruent to 1 (mod 4) and at least one prime factor is congruent to 1 (mod 4).

%C Conjecture: lim_{n->oo} a(n)/n -> 2*Pi.

%C The parameters t and G are calculated in a special Excel spreadsheet. This gives t and G for arbitrarily chosen exponent r and arbitrarily chosen shift s so that the mean value of (f(n) - a(n)/n)^2 is minimal. By changing r and s step by step I optimized the minimum of (f(n) - a(n)/n)^2.

%C Here G = limit of a(n)/n and it is less than infinity for r < 0.

%C Also, G = lim_{n->oo} A020882(n)/n, which is not only true for hypotenuses but also for odd legs of primitive Pythagorean triangles such that all prime factors of k are congruent to 1 (mod 4) and at least one prime factor is congruent to 1 (mod 4)!.

%D Lothar Selle, Kleines Handbuch Pythagoreische Zahlentripel, Books on Demand, 3rd impression 2022, chapter 2.3.2, see chapter 2.3.10 for identity of lim_{n->oo} A020882(n)/n.

%e 20 is a term and is listed twice: it is the even leg length of the Pythagorean triangles (20,21,29) and (20,99,101); GCD(20,21,29) = GCD(20,99,101) = 1, so they are primitive; and 20 = 2^2 * 5 has no odd prime factors p that are not congruent to 1 (mod 4).

%e 4 is not a term: it is the even leg length of the primitive Pythagorean triangle (3,4,5), but 4 = 2^2 has no odd prime factors.

%e 12 is not a term: it is the even leg length of the primitive Pythagorean triangle (5,12,13), but 12 = 2^2 * 3 has an odd prime factor (3) that is not congruent to 1 (mod 4).

%Y Cf. A020882.

%K nonn

%O 1,1

%A _Lothar Selle_, Jun 05 2022