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A328972 Denominator of the fraction (hypotenuse - difference of legs) / (sum of legs - hypotenuse) of the n-th primitive Pythagorean triangle.

%I A328972 #25 Nov 18 2019 07:19:52
%S A328972 1,2,3,3,3,5,4,5,5,4,7,5,7,6,5,4,9,7,7,9,7,11,8,7,6,5,11,9,9,8,7,13,6,
%T A328972 11,9,10,13,8,11,15,13,11,10,9,11,8,15,7,13,12,11,11,17,9,13,8,17,13,
%U A328972 11,15,11,10,13,19,17,14,8,13,12,11,19,13,17,10,9,15,14,21,13
%N A328972 Denominator of the fraction (hypotenuse - difference of legs) / (sum of legs - hypotenuse) of the n-th primitive Pythagorean triangle.
%C A328972 A328971(n) / a(n) should contain all reduced fractions between 1 and sqrt(2) +  1 without duplicates.
%C A328972 A328971(n) (numerators) is built from the difference between the length of the hypotenuse (A020882) and the difference between the two legs (A120682) of the n-th primitive Pythagorean triangle.
%C A328972 a(n) is built from the difference between the sum of the length of the legs (A120681) and the hypotenuse of the n-th primitive Pythagorean triangle.
%C A328972 Then both numbers are divided by their GCD to get the reduced fraction.
%C A328972 All primitive Pythagorean triangles are sorted first on hypotenuse, then on long leg.
%H A328972 S. Brunner, <a href="/A328972/b328972.txt">Table of n, a(n) for n = 1..10000</a>
%H A328972 S. Brunner, <a href="https://pastebin.com/6BbfTGAh">List for n = 0..5000 together with the primitive Pythagorean triangles </a>
%e A328972 For n=13 we need the 13th primitive Pythagorean triangle:
%e A328972 36,77,85
%e A328972 ^  ^     We calculate the difference between the two small numbers: 77-36=41.
%e A328972       ^  To get our numerator we subtract 41 from the hypotenuse length: 85-41=44.
%e A328972 ^  ^     Then we calculate the sum of the two small numbers: 36+77=113.
%e A328972       ^  We subtract 85 from this sum to get the denominator: 113-85=28.
%e A328972 This gives us the fraction 44/28 and in reduced form 11/7.
%Y A328972 Numerators: A328971.
%K A328972 frac,nonn
%O A328972 1,2
%A A328972 _S. Brunner_, Nov 01 2019