A354571 Ordered even leg lengths k (listed with multiplicity) of primitive Pythagorean triangles such that all odd prime factors of k are congruent to 3 (mod 4) and at least one prime factor is odd.

12, 12, 24, 24, 28, 28, 36, 36, 44, 44, 48, 48, 56, 56, 72, 72, 76, 76, 84, 84, 84, 84, 88, 88, 92, 92, 96, 96, 108, 108, 112, 112, 124, 124, 132, 132, 132, 132, 144, 144, 152, 152, 168, 168, 168, 168, 172, 172, 176, 176, 184, 184, 188, 188, 192, 192, 196, 196

Offset

1,1

Comments

Conjecture: lim_{n->oo} a(n)/n = Pi. Also, lim_{n->oo} A354570(n)/n = Pi.

References

Lothar Selle, Kleines Handbuch Pythagoreische Zahlentripel, Books on Demand, 4th impression 2022, chapter 2.3.3, see chapter 2.3.10 for identity of lim_{n->oo} A354570(n)/n.

Links

Table of n, a(n) for n=1..58.

Example

12 is a term and is listed twice: it is the even leg length of the Pythagorean triangles (5,12,13) and (12,35,37); GCD(5,12,13) = GCD(12,35,37) = 1, so they are primitive; and 12 = 2^2 * 3 has no odd prime factors p that are not congruent to 3 (mod 4).
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.
20 is not a term: it is the even leg length of the primitive Pythagorean triangles (20,21,29) and (20,99,101), but 20 = 2^2 * 5 has an odd prime factor (5) that is not congruent to 3 (mod 4).

Crossrefs

Cf. A354570.

Sequence in context: A260526 A040133 A092538 * A309772 A335778 A022346

Adjacent sequences: A354568 A354569 A354570 * A354572 A354573 A354574

Keyword

nonn

Author

Lothar Selle, Jun 04 2022

Status

approved