|
| |
|
|
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.
|
|
3
|
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
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
|
|
|
|
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)
(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
|
|
|
|
KEYWORD
|
easy,nice,nonn
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
