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

1,13

Comments

D. N. Lehmer has proved that the asymptotic density of a(n) is a(n)/n = 1/(2*Pi) = 0.1591549...

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.

Formula

Essentially partial sums of A024362.

Example

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 + ...

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

Prog

(Haskell)
a156685 n = a156685_list !! (n-1)
a156685_list = scanl1 (+) a024362_list  -- Reinhard Zumkeller, Dec 02 2012
(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

Crossrefs

Cf. A008846, A020882, A024409, A024362, A224921.

Sequence in context: A324965 A072749 A066490 * A124230 A330780 A090973

Adjacent sequences: A156682 A156683 A156684 * A156686 A156687 A156688

Keyword

easy,nice,nonn

Author

Ant King, Feb 17 2009

Status

approved