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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

User: Lothar Selle

Lothar Selle's wiki page.

Lothar Selle has authored 3 sequences.

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.

Original entry on oeis.org

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

Views

Author

Lothar Selle, Jun 04 2022

Keywords

Comments

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

Examples

			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).

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.

Crossrefs

Cf. A354570.

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).

Original entry on oeis.org

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

Views

Author

Lothar Selle, Jun 03 2022

Keywords

Comments

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

Examples

			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)).

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.

Crossrefs

Intersection of A004614 and A120890.
Cf. A354571.

A354569 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).

Original entry on oeis.org

20, 20, 40, 40, 52, 52, 68, 68, 80, 80, 100, 100, 104, 104, 116, 116, 136, 136, 148, 148, 160, 160, 164, 164, 200, 200, 208, 208, 212, 212, 232, 232, 244, 244, 260, 260, 260, 260, 272, 272, 292, 292, 296, 296, 320, 320, 328, 328, 340, 340, 340, 340, 356, 356
Offset: 1

Views

Author

Lothar Selle, Jun 05 2022

Keywords

Comments

Conjecture: lim_{n->oo} a(n)/n -> 2*Pi.
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.
Here G = limit of a(n)/n and it is less than infinity for r < 0.
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)!.

Examples

			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).
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.
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).

References

  • 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.

Crossrefs

Cf. A020882.

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