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A354570 Ordered odd leg lengths k (listed with multiplicity) of primitive Pythagorean triangles such that all prime factors of k are congruent to 3 (mod 4). 1

%I #54 Aug 28 2022 08:41:46

%S 3,7,9,11,19,21,21,23,27,31,33,33,43,47,49,57,57,59,63,63,67,69,69,71,

%T 77,77,79,81,83,93,93,99,99,103,107,121,127,129,129,131,133,133,139,

%U 141,141,147,147,151,161,161,163,167,171,171,177,177,179,189,189,191

%N Ordered odd leg lengths k (listed with multiplicity) of primitive Pythagorean triangles such that all prime factors of k are congruent to 3 (mod 4).

%C Conjecture: lim_{n->oo} a(n)/n = Pi. Also, lim_{n->oo} A354571(n)/n = Pi.

%D Lothar Selle, Kleines Handbuch Pythagoreische Zahlentripel, Books on Demand, 4th impression 2022, chapter 2.2.1; see chapter 2.3.10 for identity of lim_(n->oo) A354571(n)/n.

%e 3 is a term: 3^2 + 4^2 = 5^2, so the triangle with sides (3,4,5) is a Pythagorean triangle; GCD(3,4,5) = 1, so it is primitive; and the odd leg length, 3, has no prime factors p that are not congruent to 3 (mod 4).

%e 5 is not a term: it is the odd leg length of the primitive Pythagorean triangle (5,12,13), but 5 (a prime) == 1 (mod 4).

%e 21 (whose prime factors are 3 and 7, both of which are congruent to 3 (mod 4)) is listed twice because it is the odd leg length of two primitive Pythagorean triangles ((20,21,29) and (21,220,221)).

%Y Intersection of A004614 and A120890.

%Y Cf. A354571.

%K nonn

%O 1,1

%A _Lothar Selle_, Jun 03 2022

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Last modified February 27 16:21 EST 2024. Contains 370378 sequences. (Running on oeis4.)