login
This site is supported by donations to The OEIS Foundation.

 

Logo

Annual Appeal: Please make a donation (tax deductible in USA) to keep the OEIS running. Over 5000 articles have referenced us, often saying "we discovered this result with the help of the OEIS".

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A046079 Number of Pythagorean triangles with leg n. 21
0, 0, 1, 1, 1, 1, 1, 2, 2, 1, 1, 4, 1, 1, 4, 3, 1, 2, 1, 4, 4, 1, 1, 7, 2, 1, 3, 4, 1, 4, 1, 4, 4, 1, 4, 7, 1, 1, 4, 7, 1, 4, 1, 4, 7, 1, 1, 10, 2, 2, 4, 4, 1, 3, 4, 7, 4, 1, 1, 13, 1, 1, 7, 5, 4, 4, 1, 4, 4, 4, 1, 12, 1, 1, 7, 4, 4, 4, 1, 10, 4, 1, 1, 13, 4, 1, 4, 7, 1, 7, 4, 4, 4, 1, 4, 13, 1, 2, 7 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,8

COMMENTS

Number of ways in which n can be the leg (other than the hypotenuse) of a primitive or nonprimitive right triangle.

Number of ways that 2/n can be written as a sum of exactly two distinct unit fractions. For every solution to 2/n = 1/x + 1/y, x < y, the Pythagorean triple is (n, y-x, x+y-n). - T. D. Noe, Sep 11 2002

For n>2, the positions of the ones in this sequence correspond to the prime numbers and their doubles, A001751 - Ant King, Jan 29 2011

Let L = length of longest leg, H = hypotenuse. For odd n: L =(n^2-1)/2 and H = L+1.  For even n,  L = (n^2-4)/4 and H = L+2. - Richard R. Forberg, May 31 2013

REFERENCES

A. Beiler, Recreations in the Theory of Numbers. New York: Dover, pp. 116-117, 1966.

LINKS

Table of n, a(n) for n=1..99.

H. Becker, Pythagorean triples triplets in JavaScript

Ron Knott, Pythagorean Triples and Online Calculators

Project Euler, Problem 176: Rectangular triangles that share a cathetus.

F. Richman, Pythagorean Triples

A. Tripathi, On Pythagorean triples containing a fixed integer, Fib. Q., 46/47 (2008/2009), 331-340. See Theorem 6.

Eric Weisstein's World of Mathematics, Pythagorean Triple

FORMULA

For odd n, a(n) = A018892(n) - 1.

Let n = (2^a0)*(p1^a1)*...*(pk^ak). Then a(n) = [(2*a0 - 1)*(2*a1 + 1)*(2*a2 + 1)*(2*a3 + 1)*...*(2*ak + 1) - 1]/2. Note that if there is no a0 term, i.e. if n is odd, then the first term is simply omitted. - Temple Keller (temple.keller(AT)gmail.com), Jan 05 2008

For odd n, a(n) = (tau(n^2) - 1) / 2; for even n, a(n) = (tau((n / 2)^2) - 1) / 2. - Amber Hu (hupo001(AT)gmail.com), Jan 23 2008

MATHEMATICA

f[n_] := (DivisorSigma[0, If[OddQ[n], n, n / 2]^2] - 1) / 2; Table[f[i], {i, 100}] (* Amber Hu (hupo001(AT)gmail.com), Jan 23 2008 *)

PROG

(Sage) def A046079(n) : return (number_of_divisors(n^2 if n%2==1 else n^2/4) - 1) // 2 # Eric M. Schmidt, Jan 26 2013

CROSSREFS

Cf. A046080, A046081, A001227, A018892, A024361, A024362, A024363.

Sequence in context: A078692 A273432 A033151 * A279104 A165509 A100996

Adjacent sequences:  A046076 A046077 A046078 * A046080 A046081 A046082

KEYWORD

nonn

AUTHOR

Eric W. Weisstein

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified December 8 11:27 EST 2016. Contains 278939 sequences.