A156685 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.
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, 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, 11, 11, 11, 11, 11, 12, 12, 12
Offset: 1
Examples
There is one primitive Pythagorean triple with a hypotenuse less than or equal to 7 -- (3,4,5) -- hence a(7)=1. G.f. = x^5 + x^6 + x^7 + x^8 + x^9 + x^10 + x^11 + x^12 + 2*x^13 + 2*x^14 + ...
References
- Lehmer, Derrick Norman; Asymptotic Evaluation of Certain Totient Sums, American Journal of Mathematics, Vol. 22, No. 4, (Oct. 1900), pp. 293-335.
Links
- Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
- Ron Knott, Right-angled Triangles and Pythagoras' Theorem
- Ramin Takloo-Bighash, How many Pythagorean triples are there?, A Pythagorean Introduction to Number Theory, Undergraduate Texts in Mathematics, Springer, Cham, 2018, 211-226.
Programs
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Haskell
a156685 n = a156685_list !! (n-1) a156685_list = scanl1 (+) a024362_list -- Reinhard Zumkeller, Dec 02 2012
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Mathematica
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 -
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
Formula
Essentially partial sums of A024362.
Comments