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%I #22 Feb 04 2020 12:55:33
%S 0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,
%T 5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,9,9,9,9,11,11,11,
%U 11,11,11,11,11,12,12,12
%N Number of primitive Pythagorean triples A^2 + B^2 = C^2 with 0 < A < B < C and gcd(A,B)=1 that have a hypotenuse C that is less than or equal to n.
%C D. N. Lehmer has proved that the asymptotic density of a(n) is a(n)/n = 1/(2*Pi) = 0.1591549...
%D Lehmer, Derrick Norman; Asymptotic Evaluation of Certain Totient Sums, American Journal of Mathematics, Vol. 22, No. 4, (Oct. 1900), pp. 293-335.
%H Reinhard Zumkeller, <a href="/A156685/b156685.txt">Table of n, a(n) for n = 1..10000</a>
%H Ron Knott, <a href="http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Pythag/pythag.html">Right-angled Triangles and Pythagoras' Theorem</a>
%H Ramin Takloo-Bighash, <a href="https://doi.org/10.1007/978-3-030-02604-2_13">How many Pythagorean triples are there?</a>, A Pythagorean Introduction to Number Theory, Undergraduate Texts in Mathematics, Springer, Cham, 2018, 211-226.
%F Essentially partial sums of A024362.
%e There is one primitive Pythagorean triple with a hypotenuse less than or equal to 7 -- (3,4,5) -- hence a(7)=1.
%e G.f. = x^5 + x^6 + x^7 + x^8 + x^9 + x^10 + x^11 + x^12 + 2*x^13 + 2*x^14 + ...
%t RightTrianglePrimitiveHypotenuses[1]:=0;RightTrianglePrimitiveHypotenuses[n_Integer?Positive]:=Module[{f=Transpose[FactorInteger[n]],a,p,mod1posn},{p,a}=f;mod1=Select[p,Mod[ #,4]==1&];If[Length[a]>Length[mod1],0,2^(Length[mod1]-1)]];RightTrianglePrimitiveHypotenuses[ # ] &/@Range[75]//Accumulate
%o (Haskell)
%o a156685 n = a156685_list !! (n-1)
%o a156685_list = scanl1 (+) a024362_list -- _Reinhard Zumkeller_, Dec 02 2012
%o (PARI) a(n)=sum(a=1,n-2,sum(b=a+1,sqrtint(n^2-a^2), gcd(a,b)==1 && issquare(a^2+b^2))) \\ _Charles R Greathouse IV_, Apr 29 2013
%Y Cf. A008846, A020882, A024409, A024362, A224921.
%K easy,nice,nonn
%O 1,13
%A _Ant King_, Feb 17 2009
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