

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).


0



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
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OFFSET

1,1


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)!.


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.


LINKS



EXAMPLE

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).


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



