Let $a_k<b_k<c_k$ be the $k$-th primitive Pythagorean triplet in ascending order of the hypotenuse $c_k$. Define
$$ l = \frac{b_1 + b_2 + b_3 + \cdots + b_k}{c_1 + c_2 + c_3 + \cdots + c_k}, \text{ } s = \frac{a_1 + a_2 + a_3 + \cdots + a_k}{c_1 + c_2 + c_3 + \cdots + c_k} $$
Question: What is the limiting value of $l$ and $s$?
The difference between this question and the related question: Part 2: Does the arithmetic mean of sides right triangles to the mean of their hypotenuse converge? is that here the triangles are in sequenced in ascending order of the hypotenuse $c_k$ where as in the related question, they are sequenced in ascending order of $r$ and $s$, and depending on the choice of sequencing, the limiting value differs.
SageMath Code
c = 1
sa = 1
sb = 1
sc = 1
f = 0
sx = 0
while(c <= 10^20):
a = c - 1
b = 3
while(a > b):
b = (c^2 - a^2)^0.5
if(b%1 == 0):
if(b <= a):
if(gcd(a,b) == 1):
f = f + 1
sa = sa + a
sb = sb + b
sc = sc + c
sx = sx + 1/c.n()
print(f,c, sa/sc.n(),sb/sc.n(),sx)
else:
break
a = a - 1
c = c + 1
from Hot Weekly Questions - Mathematics Stack Exchange
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