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Search: id:A156685
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| A156685 |
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The number of primitive Pythagorean triples A^2+B^2=C^2 with 0<A<B<C and gcd(A,B)=1 that have an hypotenuse C that is less than or equal to n. |
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+0
1
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| 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; listen)
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OFFSET
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1,13
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COMMENT
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D. N. Lehmer has proved that the asymptotic density of a(n) is a(n)/n = 1/(2pi) = 0.1591549...
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REFERENCES
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Lehmer, Derrick Norman; Asymptotic Evaluation of Certain Totient Sums, American Journal of Mathematics, Vol. 22, No. 4, (Oct. 1900), pp. 293-335.
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LINKS
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Ron Knott, Right-angled Triangles and Pythagoras' Theorem
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FORMULA
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Partial sums of A024362
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EXAMPLE
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There is one primitive Pythagorean triple with an hypotenuse less than or equal to 7 - (3,4,5) - hence a(7)=1
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MATHEMATICA
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RightTrianglePrimitiveHypotenuses[1]:=0; RightTrianglePrimitiveHypotenuses[n_Integer?Positive]:=Module[{f=Transpose[Facto\ rInteger[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
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CROSSREFS
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A008846, A020882, A024409, A024362
Sequence in context: A124156 A072749 A066490 this_sequence A124230 A090973 A076634
Adjacent sequences: A156682 A156683 A156684 this_sequence A156686 A156687 A156688
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KEYWORD
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easy,nice,nonn
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AUTHOR
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Ant King (mathstutoring(AT)ntlworld.com), Feb 17 2009
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