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).

3, 7, 9, 11, 19, 21, 21, 23, 27, 31, 33, 33, 43, 47, 49, 57, 57, 59, 63, 63, 67, 69, 69, 71, 77, 77, 79, 81, 83, 93, 93, 99, 99, 103, 107, 121, 127, 129, 129, 131, 133, 133, 139, 141, 141, 147, 147, 151, 161, 161, 163, 167, 171, 171, 177, 177, 179, 189, 189, 191

Offset

1,1

Comments

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

References

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.

Links

Table of n, a(n) for n=1..60.

Example

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).
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).
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)).

Crossrefs

Intersection of A004614 and A120890.

Cf. A354571.

Sequence in context: A156770 A088630 A129747 * A354039 A004614 A112398

Adjacent sequences: A354567 A354568 A354569 * A354571 A354572 A354573

Keyword

nonn

Author

Lothar Selle, Jun 03 2022

Status

approved